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Margaret Daly Davis 



Piero della Francesca^ 
Mathematical Treatises 

The «Trattato d'abaco» 
and «Libellus de quinque corporibus regularìbus» 




Longo Editore - Ravenna 



Questo volume è uno studio dei rapporti fra 
la teoria matematica e la pratica artistica nel Ri- 
nascimento. L'autrice esamina il caso specifico 
di Piero, l'artista oggi universalmente ammira- 
to per la perfezione geometrica delle sue opere. 
Benché il trattato sulla prospettiva di Piero 
venga discusso tutte le volte che la prospettiva 
è all'ordine del giorno, le sue opere di mate- 
matica pura, ti Trattato d'abaco e il Libellus de 
quinque corporibus regnlaribus, sono state tra- 
scurate dagli storici dell'arte. E', però, attraver- 
so questi due trattati « minori » che la capaci- 
tà matematica di Piero, celeberrimo al suo tem- 
po, viene di nuovo messa a fuoco. 
L'autrice conduce un'indagine particolareggiata 
su questi trattati, tracciandone le fonti nella 
matematica classica e tardo-medievale e studian- 
do lo sviluppo interno della matematica picria- 
na, come pure i rapporti con n trattato sulla 
prospettiva dello stesso Piero. Di seguito alla 
esposizione del contenuto delle due opere, il la- 
voro analizza le implicazioni delle idee di Pie- 
ro e la loro diffusione nella teorìa pratica delle 
arti figurative e dell'architettura. 



MARGARET DALY DAVIS ha studiato sto- 
ria dell'arte negli Stati Uniti e in Europa. Si è 
laureata alia New York University e ha conse- 
guito il dottorato di ricerca all'Università del 
Nordi Carolina a Chapel Hill, Da tempo vive 
a Firenze dove prosegue le ricerche sulla trat- 
tatistica rinascimentale e l'arte italiana. 



In copertina: ritratto di Luca Padoli, Stimma de 
arithmetica, fol. lllv. 



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Piero della Francesca' 

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1. 

Margaret Daly Davis 

Piero della Francesca' s Mathematical Treatìses: 

The « Trattato d'abaco » and « Libellus 

de quinque corporibus regularibus » 




Speculimi Àrdimi 

Collana diretta da Aldo Scaglione 
della University of North Carolina, Chapel Hill 



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Margaret Daly Davis 



Piero della Francesca's 
Mathematical Treatises 

The «Trattato d'abaco» 
and «Libellus de quinque corporibus regularibus » 




LONGO EDITORE - RAVENNA 



RIASSUNTO 

1. 
Artisti e matematici: studi aritmetici, geometrici e prospettici 



In tutta la critica moderna sull'arte di Piero della France- 
sca, l'aspetto geometrico della sua opera, quell'astratta preci- 
sione quasi matematica della sua pittura, viene sempre presa 
in considerazione. Per il Longhi è « come se ogni cosa do- 
vesse diventare uno dei cinque corpi regolari per servire vera- 
mente alla costruzione prospettica, che è costruzione centrale 
assestata su piani convergenti ad un asse ideale, in quanto ci 
vuol far risentire lo spazio come volume regolare per mezzo 
dell'appostamento in esso di altri volumi regolari... » E già il 
Vasari ricorda che nelle cose di geometria e prospettiva Piero 
« non fu inferiore a ninno ne' tempi suoi, né forse che sia stato 
in altri tempi giammai; come ne dimostrano tutte l'opere sue 
piene di prospettive ». 

Quindi è strano che i due trattati matematici di Piero, 
il Trattato d'abaco e il Libellus de quinque corporihus regu- 
laribus, non siano stati studiati a fondo, malgrado vengano 
sempre menzionati come citazioni d'obbligo nel gran numero 
di pubblicazioni che riguardano la sua produzione artistica. La 
situazione è forse comprensibile poiché, a differenza della sua 
Prospectiva pingendi, i trattati minori di Piero, appunto quelli 
matematici, non sembrano a prima vista avere una relazione 
diretta e trasparente con le arti figurative. Questo lavoro mira 
a mettere in evidenza alcuni di tali rapporti e parte da un at- 



Vili Capitolo primo 

tento esame dell'Abaco e del Libellus, cercando di chiarirne 
i contenuti, le fonti e il significato storico, anche nello stretto 
senso matematico. 

Al suo tempo gl'interessi matematici di Piero erano in sé 
del tutto normali per un artista, pur ammettendo che nel suo 
caso erano altamente sviluppati. Il Rinascimento visse una 
fiorente collaborazione fra le belle arti e le scienze matemati- 
che di aritmetica e geometria, una connessione già evidente nel 
Trecento e nel Quattrocento. I molteplici rapporti fra i mate- 
matici e gli artisti del tempo vengono brevemente esaminati 
in alcuni casi tipici, specialmente nel campo della trattatistica, 
ad esempio, I commentari del Ghiberti, l'amicizia fra il Bru- 
nelleschi e il matematico Paolo Toscanelli, le relazioni di Luca 
Pacioli con Leonardo da Vinci, ed alcune situazioni venete. 
Anche se l'attività matematica di Piero va inserita nel contesto 
della viva e tradizionale sodalità fra artisti e matematici, essa 
fa parte del più generale risveglio dell'antichità classica, tro- 
vando un fondamento non soltanto in Euclide, ma anche in 
Archimede, Tolomeo, Vitruvio e Plinio. 

Il primo trattato pieriano, il Trattato dell'abaco nella Bi- 
blioteca Laurenziana (Ashb. 280) appartiene a un genere di 
testo matematico che, in principio, serviva ad insegnare la 
matematica mercantile. Con il passare del tempo fu aggiunta 
una sezione geometrica, e così gli abbachi vennero utilizzati 
come uno strumento più generale nell'insegnamento matema- 
tico, anche nel campo artistico-tecnico. Con un'impostazione 
molto meno pratica del libro dell'abbaco comune, V Abaco di 
Piero tratta la materia in un modo quasi puramente matema- 
tico, e passa oltre i confini dell'abbaco normale per considera- 
re problemi sofisticati di geometria stereometrica, aprendo co- 
sì la strada al Libellus sui corpi regolari. 

A differenza dell' Abaco, il Libellus de quinque corporibus 
regularibus non rientrava in un genere di trattato già stabilito. 
Per la prima volta i cinque corpi regolari — il tetraedro, il 
cubo, l'ottaedro, l'icasoedro, e il dodecaedro — vengono trat- 



Artisti è matematici IX 

tati in un libro affine agl'interessi degli artisti, e in seguito il 
tema verrà riproposto in manuali e trattati più prettamente 
artistici. Il trattato di Piero si ispira rigorosamente agli Ele- 
menti di Euclide, mentre le implicazioni cosmologiche dei cor- 
pi regolari di Platone, il quale attribuisce ad essi un profondo 
significato filosofico, vengono tralasciate. 

Il Libellus si divide in quattro parti. La prima si occupa 
di figure geometriche bi-dimensionali; la seconda, dell'iscri- 
zione di corpi stereometrici nella sfera; la terza, dell'iscri- 
zione di un corpo regolare in un altro; e l'ultima, di corpi geo- 
metrici irregolari. 



2. 

Il « Trattato d'Abaco »: 
algebra e geometria 



1 . La provenienza del manoscritto dell' « Abaco » e il suo 
contenuto algebrico 

V Abaco di Piero, un manuale di matematica mercantile, 
fu commissionato da un Pichi di Borgo Sansepolcro, un mer- 
cante, per quanto sembra, che aveva un interesse fuori del 
comune per questioni matematiche. Identificato per primo da 
Girolamo Mancini nel 1917 come opera di Piero, il manoscrit- 
to dell'Abaco si trovava nella collezione Ashburnham di Lon- 
dra, in gran parte proveniente dalla collezione di Guglielmo 
Libri, bibliotecario, storico della matematica e appassionato 
collezionista di libri e manoscritti antichi. Difatti V Abaco lau- 
renziano faceva parte della collezione Libri (verosimilmente 
fu portato via da Sansepolcro soltanto nell'800, forse dallo 
stesso Libri, una figura dai tratti oscuri). La prova di tale pro- 
venienza è semplice. Nel 1840 a Parigi il Libri stampò un 
lungo estratto dell' Abaco (che allora gli apparteneva) nel suo 
Histoire de la mathématique en Italie, identificandolo erronea- 
mente con un'opera toscana del '300. Più acuta l'analisi del 
Libri sul contenuto matematico del testo, in cui troviamo il 
tentativo di risolvere equazioni algebriche superiori al secondo 
grado, una soluzione realizzata soltanto nel primo '500 e im- 
portantissima per lo sviluppo della scienza algebrica moderna. 



Il « Trattato d'abaco » XI 

Le origini di tali ricerche da parte di Piero, si individuano nel 
matematico duecentesco Leonardo Pisano, detto il Fibonacci. 
In questo ramo della matematica le ricerche di Piero, anche 
se non coronate da pieno successo, conseguirono un buon ri- 
sultato, tale da metterlo all'avanguardia dei matematici del 
suo tempo. 



2. Il contenuto geometrico dell' « Abaco » 

La parte geometrica dell'Abaco, in un certo senso, è la 
chiave per tutta la trattatistica di Piero. Sono gli esercizi geo- 
metrici a determinare gli effettivi collegamenti con il suo Li- 
bellus e la sua Prospectiva pingendi. Mentre il libro dell'ab- 
baco, tipico del '400, è soprattutto un manuale pratico e trat- 
ta, per esempio, della misurazione di oggetti di carattere geo- 
metrico tratti dalla vita quotidiana, V Abaco di Piero invece è 
concepito in una maniera tutta diversa. La sua ideazione è 
severa, chiara, precisa ed esatta, staccata nettamente dalla real- 
tà contingente. Su un piano esclusivamente teorico viene trat- 
tata la misurazione di poligoni e poliedri di carattere geome- 
trico ed astratto. 



3. La « sezione aurea » e i cinque corpi regolari nelV« Aba- 
co » di Piero 

Nella trattazione di questi poligoni e poliedri due temi 
ci colpiscono: la « sezione aurea » e i corpi regolari. Anche 
qui le fonti delle ricerche di Piero sono da identificarsi nel 
Fibonacci e in Euclide. È importante riconoscere che Piero 
tratta la proporzione « avente il mecco et doi stremi » senza 
mai accennare al carattere particolare di essa. Questa propor- 
zione della matematica antica era indispensabile alla costru- 
zione euclidea del pentagono e conseguentemente a quella del 



XII Capitolo secondo 

dodecaedro, e trovava il suo naturale sbocco nella dottrina dei 
cinque corpi regolari, così importante per i metodi costruttivi 
di Piero. Più tardi, nell'opera del Pacioli, la « sezione aurea » 
viene denominata « divina » e acquista quegli ipertoni misti- 
co-speculativi tipici della sua applicazione alla teoria dell'arte. 
Tali elementi però mancano del tutto nel puro e rigoroso trat- 
tato matematico di Piero. Anche la trattazione dei corpi re- 
golari procede su basi euclidee, e qui Piero getta le basi per 
il Libellus de quinque corporibus regularibus, un'opera an- 
cora più prettamente euclidea. 



4. L'« Abaco » e la « Prospettiva pingendi » 

Anche i tre libri del trattato di Piero, De prospectiva pin- 
gendi, posano su fondamenti euclidei. Benché sia apparente- 
mente un'opera di matematica pura, Y Abaco ha pure una sua 
attinenza alle arti figurative. È precisamente la misurazione 
delle superna di corpi geometrici che forma la base della scien- 
za prospettica. « Prospectiva » si chiama anche « Commensu- 
ratio » e, nel linguaggio pieriano, questa commensurazione 
« diciamo essere essi profili et contorni proportionalmente po- 
sti nei luoghi loro ». Il primo libro della Prospectiva tratta 
dei punti, delle linee e dei piani, e le definizioni di un abbaco 
vengono riformulate per l'artista. Qui si ricorda appunto la 
opera Elementi di pittura dell'Alberti, ma in Piero il discorso 
è condotto matematicamente fin nei minimi particolari. In con- 
fronto l'opera dell'Alberti sembra un lavoro da dilettante. 

Il secondo libro della Prospectiva pingendi tratta dei cor- 
pi stereometrici e della loro costruzione. Nell'introduzione la 
relazione con V Abaco si chiarisce: « Corpo ha in sé tre dimen- 
sioni: longitudine, latitudine et altitudine; li termini suoi sono 
le superficie. I quali corpi sono de diverse forme, quale è corpo 
chubo, quale tetragono che non sono de equali lati, quale è 
tondo, quale laterato, quali piramide laterale, et quale di molti 



Il « Trattato d'abaco » ' XIII 

et diversi lati, sicommo ne le cose naturali et ancidentali se 
vede. De li quali in questo secondo [libro] intendo tractare 
de la loro degradationi, netti termini posti da l'occhio socto 
angoli compresi, facendo de alcune superficie degradate nel 
primo lor base ». 

In seguito il terzo libro tratta della realizzazione prospet- 
tica delle teste e delle strutture architettoniche. Qui il discor- 
so si avvicina ancora di più alla pittura: « Et perché la pictura 
non è se non dimostrationi de superficie et de corpi degradati 
o accresciuti nel termine, posti secondo che le cose vere vedute 
da l'occhio socto diversi angoli s' apresentano dicto termine... ». 



3. 

Il « Libellus de quinque corporibus regularibus »: 
corpi regolari e irregolari 



Il Libellus (Vat. Urb. lat. 632) fu steso fra il 1482 e il 
1492, l'anno in cui morì Piero. La dedica al giovane duca Gui- 
dobaldo da Montefeltro si basa su una formula vitruviana (cfr. 
De architectura, Lib. Ili, prefazione) ed inoltre svela i mo- 
tivi e gli scopi dell'autore nello scrivere il trattato, e in par- 
ticolare il suo desiderio di collocarlo accanto all'altro tratta- 
to, il De prospettiva pingendi, che era stato dedicato al pa- 
dre, ormai morto, del nuovo duca Guidobaldo, chiara testi- 
monianza di una sostanziale continuità di struttura e tema 
fra i due trattati. Il successo più notevole di Piero, nel nuo- 
vo trattato, fu l'applicazione di princìpi aritmetici ai teo- 
remi geometrici di Euclide, un risultato già preparato nel 
suo Trattato d'abaco, e poi proseguito con nuove riflessio- 
ni sul libro XV degli Elementi di Euclide. A questo punto 
vengono esposti i molteplici rapporti intercorsi fra il Libel- 
lus e V Abaco, tanto sul piano del confronto dei testi, quan- 
to a livello contenutistico. 



1. Corpi regolari: il « Libellus », parte I-I II 

Più della metà degli esercizi del Libellus (I-III) si tro- 
vano già nella parte geometrica dell'Abaco (foli. 80r-120r). 



Il « Libellus de quinque corporibus regularibus » XV 

Nel Libellus, I (dove si tratta di poligoni bidimensionali), 
33 dei 55 esercizi si ripetono dall'Abaco (nn. 3, 4, 8, 11, 
14-25, 28-30, 34-40, 42-47, 49). Di questi, 16 sono rielabo- 
rati (nn. 3, 14, 20, 24, 25, 28, 29, 35-41, 43, 44) e 17 ripor- 
tati senza cambiamenti di rilievo (nn. 4, 8, 11, 15-19, 21- 
23, 30, 34, 45-47, 49). In questa prima parte del Libellus 
si manifestano inoltre nuovi riferimenti a Euclide negli eser- 
cizi 14, 34, 39 e 46. 

Libellus, II, insegna la misurazione dei corpi regolari po- 
sti nella sfera. Dei 37 esercizi, 26 sono già nell'Abaco, 
14 sono rielaborati (nn. 4, 6, 11, 12, 14-16, 18, 21, 26-28, 
30, 35), e 12 sostanzialmente inalterati (nn. 5, 7-10, 13, 29, 
31-34, 36). 

Nel Libellus, III, iniziano le novità, specialmente nei pri- 
mi esercizi (1-13), che riflettono una connessione profonda 
con il libro XV degli Elementi di Euclide. I problemi succes- 
sivi, sulla sfera (14-29), sono quelli promessi da Piero nella 
sua introduzione al Libellus. Di questi 16 esercizi sulla sfera, 
13 sono già nei' Abaco (nn. 15-17, 19-25, 27-29), 5 in una 
forma quasi identica (nn. 17, 23-25, 28) e 8 rielaborati (nn. 
15, 16, 19-22, 27, 29), a volte con soluzioni alternative. Al- 
tri problemi dimostrano nuove utilizzazioni di Euclide (nn. 
23, 25, 27, 28) e di Archimede (nn. 15, 16, 20, 23). 

Bisogna riconoscere, però, che nel riproporre i problemi 
dell'Abaco, Piero ridefinisce le soluzioni, sia in termini di 
metodo matematico, sia in termini di esattezza numerica, con 
il risultato di un testo molto più chiaro e conciso. Si ve- 
dano, ad esempio, nel Libellus, gli esercizi I, 28 e II, 30, con- 
frontandoli con i corrispondenti problemi nell'Abaco. Tut- 
tavia la maggior precisione del Libellus, dal punto di vista 
stilistico, sembra largamente Movuta all'intervento del suo tra- 
duttore in latino. 

Le fonti delle novità contenute nel Libellus, III, si iden- 
tificano principalmente in nuovi studi fatti sul libro XV degli 



XVI Capitolo terzo 

Elementi di Euclide. Gli esercizi 1-13 trattano della misura- 
zione di un corpo regolare posto in un altro, materia non in- 
contrata nell'Abaco, e costituiscono quindi un'ulteriore tappa 
nello sviluppo matematico dell'artista. 

Difatti all'inizio di questa « tertia pars », Piero effettua 
una operazione di quantificazione dalle proporzioni geometri- 
che fisse esistenti fra i corpi regolari, l'uno iscritto nell'altro, 
derivandone numerosi esercizi matematici. Quasi tutti i pro- 
blemi hanno una loro derivazione immediata da Euclide, cor- 
rispondenza specifica in una tavola a pagina 57. 



2. Corpi irregolari-, il « LibeUus de quinque corporibus », 
Parte IV 

Nel LibeUus, IV, si passa alla considerazione di corpi irre- 
golari. Gli esercizi per la maggior parte sono di natura pret- 
tamente matematica e di carattere assai astratto. Tuttavia qual- 
che eccezione si collega a temi architettonici e, nell'insieme, 
la materia ha una sua implicita componente prospettivistica. 
Come in precedenza, si possono individuare esercizi ricavati 
dall'Abaco (nn. 2-9, 12-15, 18). 

Il LibeUus, IV, inizia con la complessa figura di un corpo 
di 72 facce: 24 triangoli e 48 poligoni di quattro lati. Le 
implicazioni architettoniche di tale figura sono sviluppate e- 
splicitamente nella Divina proporzione di Luca Pacioli, che la 
riferisce all'emisfero e allo « antico tempio pantheon », no- 
minandola « forma assai accomodata » agli architetti nel fare 
tribune e altre volte. Tali forme diventano argomenti per pro- 
spettive architettoniche nella Prospectiva pingendi di Piero, 
dove sono anche illustrate le raffigurazioni in prospettiva di 
colonne di 8 e 16 facce, e « una volta in crociera », che, nel 
LibeUus, trovano corrispondenze in analoghi esercizi di misu- 
razione (colonna e volta, « per modum crucis »). Ancora una 
volta l'intreccio « prospettiva-misurazione di corpi irregolari » 



Il « Libellus de quìnque corporibus regularìbus » XVII 

diventa più esplicito nell'opera del Pacioli, come si vede nella 
Divina proportione. 

Forse in modo meno evidente le applicazioni alla pittura 
della misurazione di poligoni e poliedri, regolari e irregolari 
si avvertono nella Prospectiva pingendi, ma ci sono. E, nello 
stesso modo, due argomenti centrali — la « sezione aurea » e 
i corpi regolari — collegano V Abaco al Libellus. La passione 
di Piero per la misurazione esatta e la raffigurazione accurata 
delle forme solide della natura, opera come una forma unifi- 
catrice in tutti e tre i trattati. 

La conclusione del capitolo prende in esame alcuni riflessi 
di simili corpi regolari e irregolari in vari rami del disegno: 
in architettura, in scenografia e nella produzione di tarsie, una 
attività allora molto fiorente. Nelle tarsie abbiamo di fronte 
un mondo irreale di strumenti scientifici e musicali, di motivi 
sacri e secolari di natura morta, tutti messi accanto a prospet- 
tive urbane, e con aggiunti astratti corpi stereometrici, in una 
visione che sembra la materializzazione delle aspirazioni dei 
trattati pieriani. 



4. 
Applicazioni dell'arte del misurare 



Le invenzioni geometriche dell'Abaco laurenziano e del 
Libellus vaticano di Piero furono rese accessibili a un vasto 
pubblico soltanto attraverso le opere di Fra Luca Pacioli. Nel 
1494 il Pacioli pubblicò gli esercizi stereometrici dell'Abaco 
nella conclusione della sua Summa arithmetica, senza però ac- 
cennare al vero autore di essi. Più tardi, nel 1509, in modo 
del tutto simile, incluse il Libellus di Piero nella sua Divina 
proportione. Così i due trattati di Piero sui corpi regolari po- 
terono più facilmente diventare la base per ulteriori sviluppi 
nella trattatistica artistica, ciò che poi è avvenuto non soltanto 
in Italia ma anche fuori. Questo fenomeno viene esaminato 
principalmente per quanto concerne gli studi e le ricerche sul- 
la prospettiva. 

I legami fra i corpi regolari e la prospettiva, non del tutto 
evidenti, si avvertono già nella Prospectiva pingendi di Piero, 
-dove è lasciato intendere che lo studio di poligoni e poliedri geo- 
metrici costituisce il primo passo nell'apprendimento della pro- 
spettiva. Più tardi nelle pagine del Vasari leggiamo di un di- 
segno prospettico molto elaborato di Piero, « un vaso in modo 
tirato a quadri e facce, che si vede dinanzi, di dietro e dagU 
lati, il fondo e la bocca: il che è certo cosa stupenda, aven- 
do in quello sottilmente tirato ogni minuzia, e fatto scor- 
tare il girare di tutti quei circoli con molta grazia ». Questo 



Applicazione dell'arte del misurare XIX 

magnifico disegno esemplifica il profondo sapere di Piero, 
« maestro raro », che meglio di ogni altro intese « tutti t 
migliori giri tirati nei corpi regolari ». 

Nella Summa arithmetica il Pacioli elenca vari artisti con 
i quali aveva discusso la prospettiva (Gentile e Giovanni Bel- 
lini, Andrea Mantegna, Melozzo da Forlì, Signorelli, Botti- 
celli ed altri), « quali sempre con libella e circino lor opere 
proportionando a perfection mirabile conducano ». D'altron- 
de egli consiglia loro la Prospectiva pingendi, un « dignissimo 
compendio e per noi ben apreso ». Insieme alla versione ita- 
liana del Libellus sui corpi regolari di Piero, nella Divina pro- 
portione, il Pacioli presenta cinquantanove grandi silografie di 
corpi geometrici, regolari e irregolari, disegnati in prospettiva. 

La rapida diffusione delle idee e delle invenzioni di Piero 
è stata assicurata, in parte, dal particolare pubblico a cui il 
Pacioli indirizzava i suoi libri. Nella Summa egli indicò come 
destinatari: architetti, prospettivisti e intarsiatori. Nella Divi- 
na proportione, « ciascun studioso di Philosophia, Prospecti- 
va, Pictura, Sculptura, Architectura, Musica e altre Mathema- 
tice ». Di questa opera di divulgazione si trova l'eco nel ri- 
tratto del Pacioli dipinto nel 1495, adesso nel Museo di Capo- 
dimonte a Napoli. Il quadro raffigura il dotto frate occupato 
in esercitazioni euclidee sui corpi regolari, che in un certo 
senso ripropongono gli insegnamenti di Piero. 

Il Pacioli si sente come un maestro degli artisti e la sua 
opera non era certa inefficace. Difatti ci conduce, per prende- 
re in esame soltanto tre esempi, ai trattati del grande Diirer 
e di un orefice suo conterraneo, il norimberghese Wenzel Jam- 
nitzer, e, di nuovo in Italia, ai trattati di Daniele Barbaro. 

Come tutto il lavoro teorico del Dùrer, l'ultimo libro della 
sua Unterweisung der Messung si fonda su premesse italiane. 
Dedicata agli studi di prospettiva e dei poliedri, questa parte 
della Unterweisung, invece di occuparsi della misurazione ste- 
reometrica secondo il severo metodo pieriano, presenta model- 
li grafici dai quali si possono costruire in carta i cinque corpi 



XX Capitolo quarto 

regolari, nove altri corpi irregolari e la sfera. Poi il Diirer 
illustra la raffigurazione prospettica di un corpo regolare, il 
cubo, e il metodo per metterlo in luce e ombra. Tale modello 
si presta ad una generalizzazione molto ampia, poiché il Dii- 
rer scrive che, avendo la pianta, ogni cosa si potrebbe dise- 
gnare nella stessa maniera. In questo modo le applicazioni 
pratiche dei corpi geometrici diventano vastissime, almeno 
sul piano teorico. Infatti il suo libro è pervaso dal costante 
riferimento alla pratica dell'artista e nella Proportionslehre 
il Dùrer tratta perfino anatomie in termini geometrici. 

L'opera del Diirer ispirò una lunga serie di piccoli tratta- 
ti, i cosidetti Kunstbiichlein and Perspectivbuchlein tedeschi. 
Fra questi, nel 1568, apparve il libro dello Jamnitzer con un 
caratteristico titolo interminabile, la Perspectiva corporum 
regularium, das ist ein fleyssige Furiveisung wie die Fiinjf Regu- 
lierten Córper, darvon Plato in Timaeo unnd Euclides inn sein 
Elementis schreibt etc, durch einen sonderlichen newen be- 
henden unnd gerechten Weg gar kunstlich inn die Perspecti- 
va gebracbt.... Da una parte questo libro, pieno di una gran va- 
rietà di forme geometriche in magnifiche tavole di rame, rappre- 
senta il culmine delle ricerche di Piero, del Pacioli e del Dùrer, 
in quanto illustra tutti i cinque corpi regolari e in più cento- 
cinquanta varianti di essi. Dall'altra l'opera dello Jamnitzer è 
una semplificazione della tradizione dei grandi maestri. Nel te- 
sto manca ogni indicazione utile per la costruzione dei corpi 
geometrici, e non c'è niente o quasi delle teorie matematiche e 
proporzionali fondamentali ad essi relative. Qui abbiamo so- 
stanzialmente esercizi accademici di stereometria utili nello stu- 
dio della prospettiva. Così il libro dello Jamnitzer diventa un 
vero libro di modelli. Questo fatto, però, non cancellava com- 
pletamente la sua origine. E fu Egnazio Danti, biografo del 
Vignola e noto matematico, che nel 1583 scrisse di « Wence- 
slao Giannizzero Norinbergese, il quale ha messi in Prospettiva 
li corpi regolari, et altri composti, si come fece Pietro dal Bor- 
go... ». 



Applicazione dell'arte del misurare XXI 

Dopo la loro prima pubblicazione in volgare, i due trattati 
del Durer, la Unterweisung e la Proportionslehre, uscirono in 
edizione latina, e furono resi così più accessibili alla cultura ita- 
liana del Cinquecento. In quel periodo la dottrina della pro- 
spettiva fu coltivata, in senso pratico, attraverso l'opera del 
dotto commentatore di Vitruvio e patriarca di Aquileia, Danie- 
le Barbaro, la Pratica della perspettiva (Venezia 1569), opera 
divenuta celeberrima in Italia. Il Barbaro assorbì moltissimi 
concetti da Piero, ma la sua dimestichezza con il pensiero pie- 
riano sembra derivare in parte dagli scritti del Pacioli e del 
Durer. La sua Pratica della perspettiva parte da istruzioni mol- 
to particolareggiate per i poligoni e i poliedri, per giungere a 
prospettive architettoniche, un interesse scenico del tutto na- 
turale per un teorico dell'architettura. 

Meno nota della sua Pratica della perspettiva è la quasi 
omonima Pratica della prospettiva, sempre del Barbaro, ri- 
masta in manoscritto (circa 300 pagine), adesso nella Biblio- 
teca Marciana a Venezia. La Prospettiva non è altro che un 
trattato vero e proprio sui corpi regolari, e come tale la più 
esauriente trattazione mai scritta sull'argomento. Una breve 
introduzione ricapitola i princìpi basilari della prospettiva, ma 
il resto del lungo trattato è dedicato al disegno prospettico di 
corpi geometrici. Qui, come nel libro stampato, il Barbaro 
collega il tema con la raffigurazione di motivi architettonici e 
infine con la scenografia: 

Hor perche gli e necessario volendo per arti di prospettiva desi- 
gnari sopra uno quadro qualche architettura, over altra cosa che 
star debbia, come essa perspettiva richiede, intendo dir il suo 
perfetto, non dico perfetto quanto a tutta la forma: perche pre- 
supono che uno dipintor, oltre la perspettiva debba perfettamen- 
te intendere quali siano li buone forme: si di gli edificij, come 
da tutti li altre cose: volendo nel arte sua procedere secondo che 
ella richiede: ma solamente intendo per il suo perfetto haver 
cognitione che forma, che li habbia, ciò è triangolare, quadrata, 
pentagona, over di altra sorte: et come tutti li anguli di essa 
architettura, over altra cosa essendo materialmente costruita. 



XXII Capitolo quarto 

E' presentata un'immensa varietà di poliedri e nume- 
rosi esempi di un corpo iscritto in un altro, per giungere a 
corpi elevati e stellati e mazzocchi complicatissimi. Un'altra 
parte del trattato si occupa della sfera, e anche questa appar- 
tiene alla pratica dell'artista: « ... et se un pittore volesse far 
in pittura un corpo spherico, over balla, tirra col compasso un 
circolo, et quello date le sue ombre diligentemente havrà fatto 
lo detto corpo spherico ». La massima parte del trattato è dedi- 
cata al disegno prospettico dei corpi geometrici, un argomento 
pieno di difficoltà, che comunque per il Barbaro merita tutta la 
fatica che richiede. Nel suo insieme l'opera rivela il fascino dei 
corpi regolari non meno dell'opera del Pacioli, ma nel Barbaro 
mancano le speculazioni filosofiche o metafisiche che si trova- 
no nel Pacioli. I corpi geometrici vengono presentati invece 
come il vero fondamento per la corretta rappresentazione di 
spazio e volume nelle arti figurative. 



Appendice 1. 
Luca Pacioli e il «Trattato d'abaco» di Piero 



Nella vita di Piero, il Vasari scrisse: « venuto Piero in 
vecchiezza ed a morte, dopo avere scritto molti libri, maestro 
Luca [Pacioli] detto, usurpandoli per se stesso, gli fece stam- 
pare come suoi, essendogli pervenuti quelli alle mani dopo la 
morte del maestro ». L'accusa del Vasari al Pacioli di aver pla- 
giato i lavori di Piero, suscitò numerosi echi, a volte non senza 
accenti polemici, principalmente nei commenti alle varie edi- 
zioni dell'opera vasariana. Per risolvere la questione, lunga- 
mente discussa, mancava assolutamente ogni dato concreto. 
Soltanto nel 1880 il tedesco Max Jordan, avendo rintracciato 
il manoscritto del Libellus de quinque corporibus regularibus 
di Piero nella Biblioteca Vaticana, riconobbe in esso la versio- 
ne latina di un trattato italiano, intitolato Libellus in tres par- 
tiales tractatus divisus quinque corporum regularium et de- 
pendentium, pubblicato in appendice nella Divina proporzio- 
ne (1509) dal Pacioli, avendo così la conferma che il Pacioli 
si era appropriato del lavoro di Piero. Ma non si deve dimen- 
ticare che il concetto del plagio, definito in senso moderno, è 
piuttosto tardo, e che nel Rinascimento tale concetto era an- 
cora piuttosto labile. 

Alle dimostrazioni dello Jordan si può aggiungere un al- 
tro caso. L'ultimo capitolo della Summa de arithmetica, geo- 
metria, proportioni et proportionalità del Pacioli (Venezia 



XXIV Appendice prima 

1494) si intitola: Particularis tractatus circa corpora regula- 
ria et ordinaria. È un trattato stampato su una decina di grandi 
pagine che concerne la misurazione dei corpi regolari 
e delle loro varianti irregolari. Questo trattato, con una dedica 
a Guidobaldo da Montefeltro tutta sua, non rappresenta altro 
che una serie degli esercizi dell'Abaco di Piero (foli. 105r- 
120r), e cioè gli esercizi stereometrici discussi sopra nel capi- 
tolo IL 27 problemi, più della metà, sono riportati quasi alla 
lettera (nn. 3-5, 11, 14, 16, 19, 20, 25, 27-30, 32, 33, 35, 
37-44, 46, 54, 55); 10 diventano un po' più concisi (nn. 6, 9, 
10, 13, 15, 17, 21, 26, 34, 48); 6 abbreviati in modo note- 
vole (2, 7, 8, 22, 52, 56); 3 allungati per maggiore chiarezza 
(12, 45, 47); 4 parzialmente rielaborati (18, 15, 53, 53 bis); 
e soltanto cinque rivisti interamente (23, 24, 36, 49, 50). Tut- 
te queste corrispondenze sono elencate nella Tavola I. 

Più tardi nella Divina proporzione il Pacioli riferisce la 
sua intenzione di non trattare della misurazione dei corpi re- 
golari, « per h averne già composto particular tractato », indub- 
bio accenno al Particularis tractatus della Summa, e così ri- 
vendicando a sé la paternità dell'opera. Forse più importante, 
però, è riconoscere nella materia del Particularis tractatus 
una fase preliminare del Libellus de quinque corporibus re- 
gularibus di Piero. 



Appendice 2. 
I manoscritti di Piero e gli adattamenti del Pacioli 



Da un attento esame dei quattro testi relativi alla questio- 
ne Piero-Pacioli varie conclusioni risultano verosimili. Sem- 
bra evidente che Piero componesse il testo originale del suo 
Libellus de quinque corporibus regularibus in italiano,, per 
lasciarlo tradurre in latino da un umanista amico, molto pro- 
babilmente da identificarsi nella persona di Matteo da Borgo. 
A sua volta la versione italiana pubblicata dal Pacioli nel 
1509 (il Libellus in tres partiales della Divina porportione) 
rappresenta una rivolgarizzazione del Libellus vaticano di Pie- 
ro, e non la versione italiana primitiva. 

Come è stato dimostrato in precedenza una gran parte del 
Libellus vaticano di Piero era già presente nel suo Abaco. 
Ciò nonostante, ci sono divergenze di sostanziale portata fra 
le due opere, e quella più tarda sviluppa certamente una ma- 
tematica più evoluta. È impossibile che il Pacioli non si sia 
accorto di un tale sviluppo. Tuttavia nei suoi prestiti clan- 
destini il Pacioli non si preoccupò delle disparità fra i due 
testi. Prima pubblicò il trattato più maturo, e poi quello me- 
no sviluppato, invertendo così l'ordine logico dell'opera pie- 
riana. Certo il Pacioli dovette tradurre o far tradurre uno 
dei manoscritti di Piero per divulgarne il contenuto a un più 
largo pubblico, ma di pari importanza è, forse, il desiderio 
di una maggiore diffusione della propria opera letteraria per 
mezzo di una calcolata politica editoriale. 



I. 



Artists and Mathematicians: Arithmetical, Geometrica!, 
and Perspectival Studies 



Even though it is universally agreed that geometry is a 
centrai element of Piero's painting, his two mathematical 
treatises, the Trattato d'abaco and the Ubellus de quinque 
corporibus regularibus, are known only superficially, if at ali, 
to art historians, who pay them little more than the lip-ser- 
vice of a passing mention 1 . There exists no study of their 
contents, sources, and significance that places them within 
the context of Piero's own art and mathematics, as well as 
in the context of the art and mathematics of Piero's time . 
This is perhaps understandable, for, unlike the precepts con- 
tained in Piero's De prospettiva pingendi, the theorems in 



1 For Piero and Piero studies, see the comprehensive monograph by 
Eugenio Battisti, Fiero della Francesca, 2 vols. (Milan 1971). 

2 Three treatises by Piero have come down to us, the Trattato d'aba- 
co (Florence, Biblioteca Medicea Laurenziana, MS. Ashb. 280/359-291), 
the Ubellus de quinque corporibus regularibus (Rome, Biblioteca Vaticana, 
MS. Vat.Urb.lat. 632), and the De prospectiva pingendi (Parma, Biblioteca 
Palatina, MS. 1576, in Italian; Milan, Biblioteca Ambrosiana, Cod. Ambr. 
C. 307, in Latin). Ali have been published except the Latin version of 
the Prospectiva pingendi: Trattato d'abaco, ed. G. Arrighi (Pisa 1970); 
L'opera « De corporibus regularibus » di Pietro Franceschi detto della 
Francesca usurpata da Fra Luca Pacioli, ed. G. Mancini (Rome 1916); De 
prospectiva pingendi, ed. C. Winterberg (Strassburg 1899); De prospectiva 
pingendi, ed. G. Nicco Fasola, 2 vols. (Florence 1942). 



2 Chapter One 

these two works do not seem to apply directly to painting. 
Nonetheless, this lacuna impedes a comprehensive evaluation 
of Piero's artistic achievement, for the Trattato d'abaco and 
the Libellus relate to perspective study and to the concerns 
of the Renaissance painter. They initiate a specific theme which 
emerges more fully developed in later manuals for artists. 
Seen in this light, Piero's two mathematical treatises form 
the fountain head of one branch of the literature of art, the 
territory first charted in Schlosser's Kunstliteratur. 



Piero's two-sided interest in mathematics and art repre- 
sented nothing out of the ordinary. The Renaissance witnessed 
a flourishing union between the fine arts and the mathe- 
matical sciences of arithmetic and geometry. Ite first fruits 
are visible in Trecento painting, and Piero's Abaco and Li- 
bellus, as well as his Prospectiva pingendi, are manifesta- 
tions of the same union. Indeed by the mid-fifteenth century 
the collaboration of artists and mathematicians in artistic un- 
dertakings was an everyday occurrence, and, by this time, trea- 
tises by artists evidence their independent investigations of 
arithmetical and geometrical principles 3 . Subsequently, with 
the foundation of academies in the sixteenth century, this 
traditionally informai association was institutionalized, and 
mathematics became part of the curriculum artists studied 4 . 



3 For the technical and. mathematical writings of Alberti, Ghiberti, 
Filarete, Francesco di Giorgio, Piero della Francesca, Leonardo 
and Albrecht Durer, see L. Olschki, Gescbichte der neuspracblkhen 
wissenscbaftlichen Literatur (1919; rpt. Vaduz 1965), voL I. Also H. Brock- 
haus, De Sculptura von Pomponius Gauricus (Leip2ig 1886), pp. 36- 
58; M. Boskovits, « Quello ch'e dipintori oggi dicono prospettiva: Con- 
tributions to Fifteenth-Century Italian Art Theory, » Acta bistoriae artium 
Academiae Scientiarum Hungaricae, Vili (1962), 241-60; IX (1963), 139- 
62; P. Tigler, Die Architekturtheorie des Filarete (Berlin 1963), esp. pp. 
33-39, 58-65, 68, 175-77. 

4 See Olschki, Gescbichte, II, 171-99; [Girolamo Ticciati] « Sto- 



Artists and Mathematicians 3 

Stili nearly ali mathematical study by Renaissance artists 
deserves to be viewed from the vantage point of a situation 
in which artists and mathematicians studied together and 
learned together. Although the mathematicians often unde- 
niably furnished the lead, this background is as revealing for 
artists who developed their mathematical skills independently 
as for those who were trained and encouraged to formulate 
mathematical inventions by mathematician teachers. 

Ghiberti, whose Commentari document his own scientific 
interests, draws attention to Trecento artists whose paintings 
reflect theoretical concerns. He describes one follower of Giot- 
to, a certain Stefano, « egregissimo dottore », and says that 
his paintings were executed with « grandissima dottrina ». 
Among the learned Sienese masters, Ghiberti counts Ambro- 
gio Lorenzetti, « molto perito nella teorica di detta arte », 
the art of disegno 5 . And it is in Lorenzetti's Annuciation of 
1344 that the first known instance of the use of a single van- 
ishing point occurs. This invention was predicated on the- 
oretical studies, and it would certainly have earned Ambrogio 
a reputation for being learned 6 . Later, Vasari reports that 
as a youth Ambrogio dedicated himself to the study of lit- 
erature and that in his later years he sought out the company 
of educated men 7 . Owing to Ambrogio's optical accomplish- 
ments, his educated friends have primarily been identified 
as mathematicians and students of optics 8 . 



ria della Accademia del Disegno, » in P. Fanfani, Spigolatura michelan- 
giolesca (Pistoia 1876), pp. 269-72. 

5 I commentari, ed. O. Morisani (Naples 1947), pp. 34, 38. 

6 G. J. Kern, «Die Anfànge der zentralperspektivischen Konstruktion 
in der italienischen Malerei des 14. Jahrhunderts, » Mitteilungen des Kunst- 
historischen Institutes in Florenz, II (1912), 58-60; E. Panofsky, «Die 
Perspektive als symbolische Form, » Vortràge der Bibliothek Warburg, 
1924-25 (Leipzig 1927), p. 279. 

7 Vasari-Milanesi, I, 524. 

8 Kern, « Anfànge, » p. 63. 



4 Chapter One 

From Antonio Manetti's biography we learn that Brunel- 
leschi's friendship with the mathematician Paolo dal Pozzo 
Toscanelli extended over some forty years 9 . And, elsewhere 
it is recorded that Toscanelli collaborated with Leon Battista 
Alberti in joint astronomical observations 10 . Most likely, Bru- 
nelleschi, too, learned from Toscanelli and had him to thank 
for many of the principles that lay at the foundation of his 
engineering and perspectival feats ". 

Another representative figure, Luca Pacioli, was born in 
Borgo Sansepolcro, the home town of Piero, and he became 
one of the most noted fifteenth-century mathematicians. In 
what must have been a familiar occupation for a mathemati- 
cian of his day, Luca dal Borgo, as Pacioli was often called, 
taught arithmetic and geometry to the architects and stone ma- 



9 A. Manetti, Vita di Filippo Brunelleschi, ed. D. De Robertis (Mi- 
lan 1976), p. 70. 

10 See M. Curtze, « Der Briefwechsel Regiomontan's mit Giovanni 
Bianchini, Jacob von Speier und Christian Roder, » Abhandlungen zur 
Geschichte der mathematischen Wissenschaften mit Einschluss ibrer An- 
wendungen, XII41902), 263-64, for a letter from Regiomontanus to Gio- 
vanni Bianchini: « Item tempore nostro oporteret declinationem Solis 
maximam esse g 24 m 2. Nos autem (preceptor meus et ego) instrumentis 
reperimus eam g 23 m 28 fere. M. Paulum Florentinum et D. Baptistam de 
Albertis sepe audivi dicentes, se diligenter observasse et non reperisse 
maiorem g 23 m 30, que res etiam tabulas nostras, videlicet tabulam de- 
clinationis et ceteras, que supra eum fundantur, innovare persuadete » 

11 For Brunelleschi's relationship to Toscanelli see most recently S. 
Y. Edgerton, Jr., The Renaissance Rediscovery of Linear Perspective (New- 
York 1975), pp. 61-63, 121-23. Also: Vasari-Milanesi, II, 333; Olschki, 
Geschichte, I, 45. For Toscanelli's relationship to contemporary artists, see 
further: H. Brockhaus, Vber die Scbrift des Pomponius Gauricus « De 
Sculptura,» Florenz 1504 (Leipzig 1885), pp. 36-38, 37, n. 1; G. Uzielli, 
Intorno ad un passo di Giorgio Vasari relativo a Paolo dal Pozzo Tosca- 
nelli, quale maestro di Filippo Brunelleschi (Rome 1894); idem, Ricerche 
intorno a Leonardo da Vinci, 2nd ed. (Turin 1896), I, passim; idem, La 
vita e i tempi di Paolo dal Pozzo Toscanelli (Rome 1894). For Brunelle- 
schi's education, P. Sanpaolesi, « Ipotesi sulle conoscenze matematiche, 
statiche e meccaniche del Brunelleschi,» Belle arti (1951), pp. 25-54. 



Artists and Mathematicians 5 

sons of his native town. And, stimulated by their demand for 
more precise mathematical rules and methods to employ in 
architectural undertakings, he composed for them a treatise 
based largely on his independent study of Vitruvius 12 . Beyond 
this, Pacioli formed a dose relationship with Leonardo da Vin- 
ci when both were at the court of Lodovico il Moro in Milan. 
Pacioli's treatises and Leonardo's notebooks testify to their as- 
sociation. Pacioli knew Leonardo's plans for the Sforza monu- 
ment in detail, for he relates the weight and measurement of 
the colossal bronze horse 13 , and he acknowledges that Leo- 
nardo was responsible for the designs of the stereometrie bod- 
ies appended to his Divina proporzione 14 . In turn, Leonardo» 
acquired a copy of Pacioli's Summa de aritmetica, geome- 
tria, proportioni et proportionalità upon its appearance in 1494, 
and he made a note to himself in the Codice Atlantico to learn 
the multiplication of square roots from Pacioli: « Impara la 
multiplicazione delle radici da maestro Luca » 15 . 

Tuscany, the traditional home of disegno, was not the only 
province which witnessed the marriage of art and mathema- 
tics. In the Veneto, Jacopo Bellini's perspective experiments 
were certainly encouraged by the scientist Giovanni Fontana, 
who dedicated his now lost treatise on perspective to the Ve- 



12 L. Pacioli, Divina proportione (Vènice 150?), fol. 23r: «Ali suoi 
caris... discipuli e alievi... Essendo da voi più volte pregato che oltre la 
prathica de Aritmetica e Geometria datovi insiemi ancora con quelle dar 
vi volesse alcuna norma e modo a poter conseguire el vostro disiato effecto 
delarchitectura ...» 

13 Divina proportione, fol. Ir: « Gommo ladmiranda e stupenda eque- 
stre statua. La cui alteqqa dela cervice a piana terra sonno brada 12 cioè 
37 4/5 tanti dela qui presente linea .ab. e tutta la sua ennea massa olire 
circa 200000 ... » 

14 Pacioli, Divina proportione, Dedication, and fols. 22r, 28v, 30v. 
A further notice in Pacioli's « De viribus quantitatis, » Bologna, Bibl. Univ.,. 
MS. 250, fol. 237r. 

15 E. Solmi, Le fonti dei manoscritti di Leonardo da Vinci (Turin 
1908), pp. 220-21. 



6 Cbapter One 

netian painter. Earlier Fontana had studied under Biagio Pe- 
lacani, an optics theorist, who lectured at the University of 
Padua between 1377 and 1411, and Fontana himself had prob- 
ably begun writing treatises as early as 1420 16 . Stili at Venice, 
the almost Tuscan precision of Carpaccio's perspective con- 
structions have been remarked more than once, and a recent 
writer has put the matter pointedly, by asking how the Vene- 
tian painter learned the synthetic spadai perspective of Piero 
della Francesca ". A notice contained in a manuscript by Da- 
niele Barbaro, the sixteenth-century trattatista, establishes that 
the principles of perspective were taught to Vittore Carpaccio 
and Giovanni Bellini by Hieronymo Malatini, whom Piero's 
follower, Luca Pacioli, describes as a perspective master 18 . 
And Barbaro tells that he himself learned the prin- 
ciples of perspective from the mathematician Giovanni Zam- 
berti 19 , who, it happens, was the brother of Bartolomeo Zam- 
berti, the first editor and translator of Euclid from Greek into 
Latin 20 . Thus the circle turns back upon itself , and it is telling 



16 L. Thorndike, A History of Magic and Experimental Science, IV 
(New York 1934), 155-57. Pelacani's Quaestiones perspectivae (1390) was, 
it seems, owned by Paolo dal Pozzo Toscanelli. See R. Klein, « Pom- 
ponius Gauricus on Perspective,» Art Bulletin, XLIII (1961), 211. 

17 Vittore Carpaccio: catalogo della mostra, ed. P. Zampetti (Venice 
1963), p. XXXIII. 

18 D. Barbaro, « La pratica della prospettiva, » Venice, Bibl. Marciana, 
MS. It. IV, 39 (=5446), fol. 2r; L. Pacioli, Summa de aritmetica, geo- 
metria, proportioni et proportionalità (Venice 1494), Dedication. Malatini's 
name recurs in connection with an engineering project, dated 1489, for 
the river Brenta in the Codex Zichy, Budapest, Erwin Szabo Library, fol. 
199. See M. Azzi Visentini, « Riflessioni su un inedito trattato di archi- 
tettura: il codice Zichy della Biblioteca Comunale di Budapest, » Arte 
veneta, XXIX (1975), 139-45, fig. 6. 

19 D. Barbaro, La pratica della perspettiva (Venice 1569), p. 2: « Il 
caso mi portò dinanzi un Giovanni Zamberto cittadino Vinitiano, il quale 
io ho usato per guida nella pratica della Perspettiva, & ho preso da quello 
molte cose, che mi sono state utili, & di piacere. » 

20 See Zamberti's dedication of EucHd's treatise on optics in Euclidis 



Artists and Mathematicians 7 

that in ali three cases perspective procedures were transmit- 
ted not by an artist but by a specialist mathematician. In this 
respect, it is worth remembering the stili nameless mathema- 
tician who revealed the « secrets » of perspective to Albrecht 
Diirer, for these theoretical principles were so important that 
they induced Dùrer to undertake the journey from Venice to 
Bologna to learn them 21 . 

Where artists and their own mathematica! studies and 
writings are concerned, we may discover the proper context 
of Piero's treatises by examining a sample of the mathematical- 
scientific interests manifested by his fellow artists. Here we 
may begin by turning again to Diirer. From his own notes, we 
learn that Diirer purchased Zamberti's edition of Euclid's Ele- 
ments in Venice in 1507, and his continuing involvement with 
this work is not difEcult to trace throughout his treatises °. 
The importance he attached to geometry is epitomized in the 
dedication of his Unterweisung der Messung, where he writes: 

Die weyl aber die [Kunst der Messung] der recht grundt ist aller 
mallerey, hab ich mir fiirgenomen alien kùnstbegyrigen jungen, 
eyn anfang ziistellen und ursach ziigeben damit sie sich der 
messunge zirckels und richtscheyt underwinden unnd darauss die 
rechten warheyt erkennen unnd vor augen sehen mogen, damit 



megarensis philosophi platonici ... Elementorum libros XIII ..., ed. B. Zam- 
berti (Venice 1505): « Bartholomaeus Zambertus venetus Ioanni Zamberto 
Veneto frati humanissimo salutem perpetuam. » 

21 E. Panofsky, Albrecht Diirer, 3rd ed. (Princeton 1948), I, 251; 
Albrecht Dùrer 1471-1971: Ausstellung des Germanischen Nationaltnuseums, 
Nùrnberg (Munich 1971), pp. 341-42. 

22 Dùrer: Schriftlicher Nachlass, ed. H. Rupprich, I (Berlin 1956), 221. 
The Zamberti edition of Euclid's Elements, owned by Diirer, contained the 
following annotation: « Daz puch hab ich zw Venedich vm ein Dugatn kawft 
im 1507 jor. Albrecht Dùrer.» See Durer: Nachlass, II (1966) and III 
(1969), index, s.v. « Euklid, » for his references to the Elements. 



Chapter One 

sie nit alleyn zìi kùnsten begirig werden, sonder auch zu eynem 
rechten und gròsseren verstant komen mogen 23 . 



Returning to Florence, at the very beginning of the Ren- 
aissance, one encounters the third book of Ghiberti's Com- 
mentari, a work which although it treats optics and the pro- 
portions of the human figure, is often considered little more 
than a confusing jumble of quotations. A recent penetrating 
analysis of Ghiberti's third Commentario demonstrates, how- 
ever, that Ghiberti's selection of passages from medieval ap- 
tics treatises is not random, but purposeful and systematic, 
and intended to give a sdentine basis to the study and use of 
light in the arts of sculpture and glass-painting 24 . 

The most prominent trattatista following Ghiberti is Leon 
Battista Alberti, whose works are noted for their eminently 
scientific spirit. Alberti's "De' ludi mathematici, a work of piane 
geometry, presents fairly common exercises in such problems 
as the correct measurement of towers, rivers, and fields 25 . 
His Elementi di pittura is a piane and solid geometry trea- 
tise, but its mathematical definitions have been reworked for 
artists. It begins in a Euclidean fashion by defining the point: 
« El puncto dicono essere quello che nulla si possa dividere 
in parte alcuna ». For artists this is then redefined: « Puncto 
nominamo noi in pictura quella picola jnscriptione , quale nul- 
la può te essere minore ». And an outline is defined thus: 
« Lembo nominamo tuta quella circumducta descriptione fac- 
ta da linee sotilissime per quale sia l'una area divisa da l'al- 
tra ». The artist, significantly, is not instructed to draw after 



23 Underweysung der messung mit dem zirckel und richtscheyt (1525; 
facsimile rpt. Dietikon-Zùrich 1966), verso of title page. 

24 J. Gage, « Ghiberti's Third Commentary and its Background, » Apol- 
lo, XCV (1972), 364-69, esp. 368. 

25 « De' ludi matematici, » in Opere volgari di Leon Batt. Alberti, ed. A. 
Bonucci, IV (Florence 1847), 405-40. 



Artists and Mathematicians 9 

nature, but rather he is presented with abstract regular and 
irregular polygons to the end of representing three dimen- 
sionai forms in perspective. The reader is urged to pursue these 
rudiments of painting diligently until he has grasped them. 
They are, Alberti writes, striking the doublé chord of the utile 
and the ànice, no less useful than amusing 26 . 

Even more direct observations concerning the nature of 
the relationship between geometry and art are found in Al- 
bertus Della pittura. 

Piacemi il pittore sia dotto in quanto e possa in tutte l'arti li- 
berali ma imprima desidero sappi giometria ... I nostri dirozza- 
mene, da i quali si exprime tutta la perfetta absoluta arte di 
dipigniere, saranno intesi facile dal geometra, ma a chi sia ignio- 
rante in geometria né intenderà quelle né alcun'altra ragione di 
dipigniere: pertanto affermo sia necessario al pittore inprendere 
geometria 27 . 



Thus the painter must have a sdentine understanding of 
what he sees and a knowledge of geometry to reproduce it 
correctly. Thereby naturalism is given a scientific basis, and, 
too, a path is indicated whereby modem artists may surpass 
those of antiquity. 

* -k * 

While Piero's interest in mathematics was traditional for 
an artist, his treatises, from the point of view of their content 
and their inspiration, are also a function of the rebirth of clas- 



26 « Elementi di pittura, » in Opera inedita et pauca separatim impres- 
sa, ed. G. Mancini (Florence 1890), pp. 47-65. See also A. Gambuti, «Nuove 
ricerche sugli Elementa picturae, » Studi e documenti di architettura (1972), 
no. 1, 133-72; G. Arrighi, « Leon Battista Alberti e le scienze esatte, » 
Convegno internazionale indetto nel V Centenario di Leon Battista Alberti, 
Roma — Mantova — Firenze, 25-29 aprile 1972 (Rome 1974), pp. 155-212. 

27 Della pittura, ed. L. Malie (Florence 1950), pp. 103-04. 



10 Chapter One 

sical studies 28 . Piero was undoubtedly familiar with Euclid, 
Archimedes, and Ptolemy, for he mentions them and quotes 
from their works *. Moreover, his familiarity with Vitruvius 
can, not surprisingly, be demonstrated in the dedication to the 
Libellus 30 . Vitruvius' De architectura, together with Pliny 's 
Naturalis historiae, constituted the principal literary sources 
from which the Renaissance sought to rediscover the underly- 
ing prindples of ancient art. Both prescribe the knowledge of 
mathematics as a fundamental prerequisite to the practice o£ 
art. Among the many accomplishments of Vitruvius' ideal edu- 
cated architect, one was mathematics. And without the aid of 
arithmetic and geometry, according to Pliny, art will never at- 
tain perfection 31 . In this context Piero's Abaco and his Libel- 
lus must be seen as aids for the Renaissance painter who at- 
tempted to follow a programme all'antica in the pursuit of 
artistic excellence. 

It is in this context of the dose associations between artists 
and mathematicians, a union sanctioned, even prescribed by 
classical antiquity, that the emergence of Piero's treatises be- 
longs. Furthermore, for Piero the art of painting could almost 
be subsumed under the mathematical sciences. This view was 
not without opponents, for Piero himself writes that many 
painters stili fail to understand the true significance of lines 
and angles, and they consequently disapprove of perspective 32 . 



28 Battisti, Piero della Francesca, I, 457, n. 1. 

29 In the Trattato d'abaco Euclid is referred to on fols. 87v; 90r-v; 103r; 
104r-v; 106r; llOr-v; 112r-v; Ptolemy on fols. 98r and llOv; and Archi- 
medes on fol. 106v. I follow here, and in the following pages, Arrighi's 
accurate transcription and his folio numera tion. 

30 See below Chapter III, pp. 44-45. 

31 Vitruvius, De architectura (I, i, 4) and Pliny, Naturales historiae 
(XXXV, xxxvi, 76). In this connection see also L. Lanzi, Storia pittorica 
dell'Italia, I (Bassano 1795), 49-50: « Pietro della Francesca ... par che 
fosse il primo a richiamar l'uso dei Greci, che la geometria fecero servire 
alla pittura. » 

32 Piero della Francesca, De prospectiva pingendi, ed. G. Nicco 



Artists and Mathematicians 11 

In his De prospectiva pingendi Piero defines painting as a part 
of perspective, that is to say as a brandi of geometry. 

Et perché la pictura non è se non dimostrationi de superficie et 
de corpi degradati o acresciuti nel termine, posti secondo che le 
cose vere vedute da l'occhio socto diversi angoli s'apresentano 
nel dicto termine, et però che d'onni quantità una parte è sempre 
a l'ochio più propinqua che l'altra, et la più propinqua s'apre- 
senta sempre socto magiore angolo che la più remota nei termini 
assegnati, et non posendo giudicare da sé lo intellecto la loro 
mesura, cioè quanto sia la più propinqua et quanto sia la più 
remota, però dico essere necessaria la prospectiva, la quale discer- 
ne tucte le quantità proportionalmente commo vera scientia, di- 
mostrando il degradare et acrescere de onni quantità per forza 
de linee 33 . 



Painting presupposes the ability to measure solid objects 
and to represent them scientifically, thus requiring a solidly 
founded knowledge of the principles of arithmetic and geo- 
metry. Both of Piero 's minor treatises, the Trattato d'abaco 
and the Libellus de quinque corporibus regularibus, testify to 
Piero's remarkable accomplishments in this area. Pure mathe- 
matics they are, but they are more than that, for as we will 
see they are immediately connected with the artistic concerns 

of the day. 

* * * 

Piero's Trattato d'abaco belongs to a genre of mathemati- 
cal texts rarely considered by students of the arts, yet the 
abbaco perhaps represented the first important means whereby 



Fasola, p. 128: « Molti dipintori biasimano la prospectiva, perché non in- 
tendano la forza de le linee et degli angoli, che da essa se producano. » 

33 Prospectiva pingendi, pp. 128-29. See also the discussion of paint- 
ing as a practical branch of mathematics in M. Savonarola's « Libellus de 
magnificis ornamentis regiae civitatis Paduae, » in L. A. Muratori, Rerum 
Italicarum scriptores XXIV (Milan 1738), col. 1169. 



12 Chapter One 

artists became acquainted with mathematical learning. Its ini- 
tial purpose was to teach commercial mathematics to future 
merchants, but in time, with the addition of a geometry sec- 
tion (which saw successive refinements), the abbaco books 
became more ambitious in scope and suitable for a wider pub- 
lic *. 

The abbaco texts borrow extensively from one another, 
but ali are ultimately based on the Liber abbaci (1202) and 
the V radica geometriae (1220) of Leonardo Pisano, who is 
commonly called Fibonacci 35 . Singlehandedly, Fibonacci rev- 
olutionized medieval mathematics by introducing Arabie nu- 
merals and methods of computation, as he said, « to the Latin 
races in order that they no longer be ignorant in these 
things » 36 . The Arabie inventions that he introduced were con- 
densed, translated into Italian, popularized in the trattati del- 
l'abbaco, and taught in the scuole dell'abbaco of the fourteenth 
and fifteenth centuries 37 . 

An education in a scuola dell'abbaco entailed learning the 
rudiments of arithmetic and geometry. It apparently preceded 
apprenticeship in a trade, for the average age of the students, 



34 See R. Goldthwaite, « Schools and Teachers of Commercial Arith- 
metic in Renaissance Florence, » Journal of European Economie History, 
I (1972), 418-33, and, for further bibliography, p. 418, n. 1. The abbaco 
books have only recently been introduced into art historical literature: M. 
Baxandall, Painting and Experience in Fifteenth Century Italy (Oxford 
1972), pp. 86-108; D. F. Zervas, « The Trattato dell'Abbaco and Andrea 
Pisano's Design for the Fiorentine Baptistery Door, » Renaissance Quarterly, 
XXVIII (1975), 483-503. For the possible influence of Piero's Libellus de 
quinque corporibus regularibus on the works of Bronzino, particularly on the 
Havemeyer portrait, C. H. Smyth, Bronzino as Draughtsman (Locust Valley, 
N. Y. 1971), pp. 83-85. 

35 Scritti di Leonardo Pisano, matematico del secolo decimoterzo, ed. 
B. Boncompagni, 2 vols. (Rome 1857-62). 

36 E. Colerus, Piccola storia della matematica da Pitagora a Hilbert 
(Turin 1962), p. 135; M. Cantor, Vorlesungen ùber Geschichte der Mathe- 
matik, 2nd ed., II (Leipzig 1892), 5. 

37 See below Chapter II, n. 38. 



Artists and Mathematicians 13 

it seems from family chronicles, fell between nine and twelve 
years 38 . Once a geometrical section was commonly included 
in the abbaco, the texts became more systematic and purpose- 
ful, and the books then constituted an indispensable instru- 
ment in the education of aspiring artisans, artists, architects, 
and engineers. 

The unpublished codex Ashburnham 359 in the Laurentian 
Library is representative of the whole genre of abbachi. Writ- 
ten by a Fiorentine around 1475, it begins, « Inchomincio 
uno trattato d'abacho nel quale diamostro tutto quello che so 
paritene allo merchantio » 39 . The introduction then teaches 
the multiplication tables, and succeeding chapters treat frac- 
tions, proportions, division of profit, methods of bartering, 
monetary exchange between various cities in Italy, and calcula- 
tions with Fiorentine weights, measures, and currency. The 
« regole delle tre cose », or the « rule of three », the basis for 
most commercial transactions, formed one of the most impor- 
tant chapters of the trattati d'abbaco. A simple exercise of 
this kind asks, if five braccia of wool cost seven lire, what will 
three cost? Here the several elements are reduced to the sim- 
ple proportion, 5 : 7 = 3 : x. In the Ashburnham Abba- 
co 359, however, the « rule of three » exercises develop into 
abstract proportional formulas which go beyond the require- 
ments of pure mercantile exchange and become useful propor- 
tional tools for builders and architects. The anonymous author 
of Ashburnham 359 states the matter thus: 



38 For example, we learn from Bernardo Machiavelli^ Libro di ricordi, 
ed. C. Olschki (Florence 1954), pp. 31, 45, 103, 138, that his son Nicolò 
(b. 1469) began frequenting a « maestro Matteo, maestro di grammatica, » 
at age seven, in 1476. In 1479, at age ten, he began studying with Piero 
Maria, « maestro d'abaco, » and at age twelve, in 1481, presumably having 
completed the scuola dell'abbaco, he began the study of the Latin classics. 

39 Florence, Bibl. Medicea Laurenziana, MS. Ashb. 359, fol. lOr. 



14 Chapter One 

Dobbiamo sapere che quando sono de lo più numerj de quali 
il primo sia al sechondo chome el sechondo al terzo & chome el 
terzo al quarto & chosi seguendo quelli numerj si dichono nella 
proportione chontinua e similmente quando sono in quantità e 
quando sono el primo al sechondo chome el sechondo al terzo 
& chome el terzo al quarto allora quelli numerj si dichono in 
proportione chontinua di chome siano nelle proportione chonti- 
nua e non chontinua 40 . 



Two examples are given. First, 8 : 12 : 18 : 27, which 
can be reduced to the proportion 2:3,2:3,2:3(or 
fifths in musical consonances). And second, 12 : 18 : 
24 : 36, reducible to 2 : 3, 3 : 4, 2 : 3 (in musical terms, 
a fifth, a fourth, and a fifth). The last chapter of the Ash- 
burnham Abbaco 359, chapter 24, is devoted to geometrical 
exercises ih the measurement of surfaces and volumes of sol- 
ids 41 . Various abstract polygons are described and explained, 
but the most immediately geometrical exercises are practical 
ones dealing with such problems as the measurement of the 
contents of a barrel or a conical pile of grain, of the heights of 
walls, towers, and trees, and of long stretches over a land sur- 
face. 

A survey of geometrical problems in a number of abbaco 
books shows that while they are suited to the needs of future 
merchants, they also provide, to a large extent, what future 
artisans, artists, architects, and engineers needed to learn. The 
Ashburnham Abbaco 359, for instance, also teaches the de- 
termination of the height of a column and the depth of a well, 
and it concludes with the measurement of the height of the 



40 Ashb. 359, fol. 47r. 

41 Ashb. 359, fol. 142r: « Nel 29° e ultimo chapitolo di questo trattato 
si dimostra chon brevità el modo del misurare le superficie chorpi e dalchuna 
altre cagionj piacevoli. » 



Artists and Matbematicians 15 

« palagio de signori », the Palazzo Vecchio in Florence 42 . Ash- 
burnham Abbaco 518 demonstrates how to measure variously 
shaped tracts of land, pavilions, round and cubie stones, as 
well as the diameter of a column, round towers and walled for- 
tresses. It also includes problems treating the measurement of 
unbroken walls and of walls pierced with arches, the painting 
of palace facades, the roofing of a house with tiles, the facing 
of the four sides of a house with stone, and the paving of piaz- 
zas with brick 4 \ 

Another abbaco text, Ashburnham 356, instruets in meas- 
uring round vases **, and Ashburnham Abbaco 1662 explains 
the measurement of a chalice 45 . Ashburnham 1379 shows how 
to measure columns and how to lay the brick pavement of the 
floor of a large hall 46 . Pier Maria Calandri's Trattato d'abba- 
cho measures, among other things, a column, a chalice, wells, 
stones, and « la palla di S ancia Liperata », that which 



42 Ashb. 359, fol. 206r: « ... et volessi misurare quanto e alto el pa- 
lagio de signor j. » The person taking the measurements stands on the corner 
of via Vacchereccia across from the palace and works out the solution 
using similar triangles. (Cf. also MS. Acq. e doni 154, fol. 224v.) Through 
geometrical problems in measuring buildings, like this one and the following 
one by Francesco di Giorgio, together with a study of optics treatises, the ge- 
nesis of Brunelleschi's invention might be traced. Francesco di Giorgio 
Martini, La praticha di gieometria, ed. G. Arrighi (Florence 1970), p. 21: 
« Se per uno specchio l'alteza d'una torre saper vorremo, pigia lo specchio e 
ppollo in nel piano apresso alla torre, e ppoj ti muove innanzi e 'ndirietro 
tanto che ttu dirittamente veggha el' A in nel B, cioè la cima de la torre en 
el mezzo de lo specchio. E guarda quant'è la propotione de la distantia che è 
dal mezzo de lo specchio a' tuo) piedj e IV altezza che è dal pie all'occhio tuo, 
cioè dal BC al CD, chosj he dal BE a l'AE chome el BC e l mezo dello spec- 
chio a ppiè de la torre si metta. Esse altrjmente fusse, la proportione dal BA 
al CD altrjmentj sarebbe del piano a la torre. Unde mixura lo piano e pro- 
potionalo chome di sopra hera la torre. » Cf. Boskovits, « Prospettiva, » 
Vili (1962), 244; H. Siebenhùner, « Zur Entwicklung der Theorie der 
Renaissance-Perspektive, » Kunstchronik, VII (1954), 129-30. 

43 Florence, Bibl. Medicea Laurenziana, MS. Ashb. 518, fols. 181r-207r. 

44 Florence, Bibl. Medicea Laurenziana, MS. Ashb. 356, fol. 23r. 

45 Florence, Bibl. Medicea Laurenziana, MS. Ashb. 1662, fol. HOv. 

46 Florence, Bibl. Medicea Laurenziana, MS. Ashb. 1379, fols. 87r-94r. 



16 Chapter One 

crowned the lantern of S. Reparata, the Fiorentine Duomo, 
until 1601 (Figs. 1, 2, 3) 47 . 

The abbaco books of the fourteenth and fifteenth centur- 
ies have only begun to be surveyed with an eye to gauging 
their róle in the education of artists and artisans. But it is evi- 
dent from the geometrical problems outlined above that these 
texts, the basis of the curriculum of a scuola dell'abbaco, were 
suited to teach beginning architects, engineers, painters, sculp- 
tors, goldsmiths, wood carvers, and intarsia designers. Vasari, 
in fact, relates that in his first months of study, Leonardo so 
progressed « nell'abbaco » that he confounded his master with 
the difficulty of the questions he raised 48 . 

In a way Piero's Trattato d'abaco represents the culmina- 
tion of this genre of text book in the fifteenth century. Piero's 
introduction to his Abaco relates that he was asked to write 
« alcune cose de abaco necesarie a' mercatanti » 49 . Beginning 



47 P. M. Calandri, Tractato à'abbacho, ed. G. Arrighi (Pisa 1974), fol. 
219v. The diameter of the sphere crowning the lantern of the Duomo in 
Calandri's exercise is 4 braccia, which corresponds to the true measurement 
of Verrocchio's « palla. » C. Guasti, ha cupola di Santa Maria del Fiore 
(Florence 1857), p. 113: «a misura di braccia quattro, meno l'altezza d'uno 
dito grosso. » 

48 Vasari-Milanesi, IV, 18. Manetti (above n. 9), n. 52, relates con- 
cerning Brunelleschi: « Nella sua tenera età, Filippo apparò a leggere ed 
a scrivere e l'abaco, come s'usa per gli uomini da bene e per la maggiore 
parte fare a Firenze. » Similarly, conceming Bramante, Vasari (-Milanesi, IV, 
147) wrote: « Oltre il leggere e lo scrivere si esercitò grandemente nello 
abbaco. » 

49 Trattato d'abaco, fol. 3r. Although Piero did not name his patron, 
it was one from whom « / preghi suoi me sono commandamenti » (fol. 3r). 
The stemma on the opening page of the manuscript, however, has been 
identified as that of the Picchi family, relatives of Piero from Borgo San 
Sepolcro (Ministero della Pubblica Istruzione. Indici e cataloghi, Vili: 
I Codici Ashburnhamiani della Biblioteca Medicea Laurenziana di Firenze, 
I, fase. 6, ed. E. Rostagno and T. Lodi [Rome 1948], 462 ff.). Battisti 
{Piero della Francesca, I, 281), rightly dates the treatise to the years around 
1450. For Piero's relationship to various members of the Picchi family: 
Battisti, II, index, s.v. « Picchi, famiglia di Sansepolcro. » 

Although nothing is known of Piero's schooling, he carne from a mer- 



Artists and Mathematicians 17 

with fractions and the « rule of three », Piero moves quickly 
to advanced algebraic problems which, though they are stili 
dressed in mercantile garb, transcend the needs of commercial 
transactions. The algebraic methods of computation to which 
Piero aspired were, we will see, innovative in the fifteenth 
century. Only in the sixteenth century were they finally per- 
fected and brought to their logicai conclusion. In a like man- 
ner, the geometry section of Piero's Abaco is more advanced 
than those of books similar to his. With two exceptions, there 
are no practical problems of measurement at ali. Instead Piero 
deals exclusively with the measurement of abstract polygons 
and polyhedra. He divides his treatment of geometry into two 
sections, one on piane and the other on solid geometry. The 
first problems in piane geometry lay the foundation for the 
subsequent stereometrical demonstrations. Then, following the 
pattern of Euclid's Elements, Piero teaches the measurement 
of the five regular and other irregular solids inscribed in a 
sphere. 

Although in the text of his Abaco Piero never indicates 
that he considers his treatise an advanced work, a statement in 
Alberti's De' ludi matematici shows that Piero's theme is more 
difficult than that which Alberti himself treats. « Potrei in si- 
mili cose molto estendermi, ma queste per ora bastino. Se altro 
mi chiederete lo farò volentieri. Le misure de' corpi come sono 
colonne tonde, quadre, e auzze, di più f accie, speriche e simili 



cantile family. L. Coleschi, Storia della città di Sansepolcro (Città di Ca- 
stello 1886), p. 262, describes Piero's father Benedetto as a « mercante di 
panni di lana e calcolatore »; see also Battisti I, 31. For a notice of Pie- 
ro's activity as « ingegnere idraulico, » see G. Sacchetti, Sansepolcro, 
Alta Valle Tiberina (Sansepolcro 1888), p. 78. 

For a device related to that on fol. 3r of Piero's Trattato d'abaco, see 
P. M. Calandri's Tractato d'abbacho (n. 47). Piero's scroll reads PI ... 
TEXT ... SI. The three scrolls in Calandri's text (fols. 3r, 9r, and lOr), read 
respectively: TEXT ... EU ... PSQ; TEXTE ... IP ... S .. V; TEXT ... EU. 
TEXT may imply textere, 'to weave' and relate to the wool guild. 



18 Chapter One 

corpi, e le tenute de' vasi e simili sono materie più aspre a trat- 
tare pur quando a voi dilettasse potrò ricorvele » 50 . Alberti's 
Elementi di pittura testifies, however, that exactly such studies 
dealing with multi-sided polygons constituted the basis of 
many artistic exercises for painters. Moreover, the complexities 
of Piero's particular problems ensured their usefulness for ar- 
chitects, engineers, sculptors, and intarsia designers. In fact, we 
will see that the geometry section of Piero's Trattato d'abaco 
is closely related to his Prospectiva pingendi, and that it forms 
the basis of the Libellus de quinque corporibus regularibus. 

* * * 

Unlike his Abaco, Piero's book on the five regular bodies 
(the tetrahedron, cube, octahedron, icosahedron and dodeca- 
hedron) did not belong to a pre-existing genre. Instead, like 
the Prospectiva pingendi, it is originai in conception and con- 
tent. The Libellus presents for the first time a theme treated in 
many artists' manuals from the sixteenth century onwards. 
Such manuals aimed, at least in part, to teach the construction 
and perspectival drawing of the regular bodies. 

The inspiration for Piero's treatise is found in the final 
books of Euclid's Elements — Books XIII, XIV, and XV — 
which culminate in the description, the construction, and the 
analysis of the relationships existing among the five regular 
polyhedra, the only equilateral, equiangular geometrical solids 
which can be comprehended in a sphere 51 . In Plato's Timaeus 
the five regular bodies were ascribed a cosmological signifi- 
cance as the symbols of the elements. The cube was associated 



50 « Ludi, » p. 439. 

51 See The Tkirteen Books of Euclid's Elements, ed. T. L. Heath, 2nd 
ed. rev. (New York 1956), III, for books XIII and XIV and a discussion 
of Book XV. Books XIV and XV, now known not to be the work of 
Euclid, stili enjoyed his authority in the fifteenth century. For Books XIV 
and XV in the Campanus translation, which Piero used, see Euclidis Mega- 
rensis philosophi acutissimi ... Opera, ed. L. Pacioli (Venice 1509). 



Artists and Matkematicians 19 

with earth, the octahedron with air, the pyramid with fire, 
the icosahedron with water. Finally the dodecahedron, which 
approaches most closely to the sphere, was associated with 
the cosmos for God used it « for arranging the constellations 
of the whole universe » 52 . 

The topic of the regular bodies was thus a fitting one for 
a Renaissance trattatista to choose from a philosophical as well 
as from an artistic and mathematical point of view. The accu- 
rate measurement of volumes and surfaces furnishes an exer- 
cise in the elements of sound painting, and, at the same time, 
it necessitates a through knowledge of mathematics. Although 
Piero does not mention the Platonic aspect of the regular bod- 
ies, it was of current interest, and he doubtlessly knew it 
well, for the cosmic, platonic implications of the regular bodies 
are expressly outlined in the penultimate proposition of Book 
XV of Euclid's Elements 5 \ 

Piero dedicated the manuscript of his Libellus de quinque 
corporibus regularibus to Guidobaldo da Montefeltro, the son 
of Federigo da Montefeltro, Duke of Urbino, for whom 
Piero had earlier written his Prospectiva pingendi. In his ded- 



52 Plato, Timaeus, transl. with an introd. by HJD.P. Lee (Baltimore: 
Penguin, 1965), p. 75. 

53 Euclid, Opera, ed. Pacioli, Book XV, prop. 12, fol. 144r-v. Other 
treatises on the five regular bodies are: (1) Triest, Biblioteca Civica «Atti- 
lio Hortis », R. P. MS. 2-33, discussed by G. Arrighi in Francesco di 
Giorgio Martini, La praticha di gieometria, pp. 7-12, and (2) Regiomonta- 
nus ( = Johann Muller of Kònigsberg), « De quinque corporibus aequi- 
lateris, quae vulgo regularia nuncupantur, quae videlicet eorum locum im- 
pleant naturalem & quae non, contra commentatorem Aristotelis, Averro- 
em, » listed in J. G. Doppelmayr, Historische Nachricht von den Niirn- 
bergischen Mathematicis und Kunstlern (Nuremberg 1730), p. 19. Regio- 
montanus's treatise no longer exists, although there is a list of its chapters 
in a copy of another work, his « Commensurator. » See W. Blaschke and 
G. Schoppe, Regiomontanus: Commensurator (Wiesbaden 1956), pp. 474- 
75. The chapter headings are concerned with the construction of ali five 
bodies, the measurement of some of them, and the derivation of one body 
from another. 



20 Chapter One 

ication Piero asks that the Libellus be placed with his book 
on perspective in the Urbino Library M . In the event, Piero's 
wishes were carried out and the books were shelved together. 
The proof is that, since Piero's perspective treatise lacked a 
title, Veterano's fifteenth-century catalogue entry for the man- 
uscripts in the Urbino library records both works together as 
one: « 273. Vetri Burgensis pictoris Libellus de quinque cor- 
poribus regularibus, ad illustrissimum ducem Federicum et 
Guidonem filium » 55 . 

The Libellus is divided into four parts. The first treats 
piane geometrical figures; the second, solid bodies inscribed 
in a sphere; the third, problems of regular bodies placed with- 
in one another and exercises concerning spheres; the fourth, 
irregular bodies. That Piero intended the book to stand « pe- 
nes aliud nostrum de Prospectiva opusculum, quod superiori- 
bus annis edidimus » K indicates that he felt strongly the con- 
tinuity of content between the Libellus and the Prospectiva 
pingendi. And, as we will see, that the topic of the regular and 
irregular bodies became an inevitable component of the exer- 
cises in later perspective manuals supports this conclusion. 



54 De corporibus regularibus, p. 488. 

55 C. Guasti, « Inventario della libreria urbinate compilato nel secolo 
XV da Federigo Veterano, bibliotecario di Federigo I da Montefeltro, Duca 
d'Urbino, » Giornale storico degli archivi toscani, VII (1863), 55. 

56 See above n. 54. 



II. 

The « Trattato d'abaco »: 
Algebra and Geometry 



1 . The Provenance of the « Abaco » manuscript and its alge- 
braic content 

Piero della Francesca's Trattato d'abaco, like other abbaco 
books of its kind, is a handbook of arithmetic, algebra, and 
geometry, written as a guide to mercantile mathematics. To 
this fact its opening lines testify explicitly. 

Esendo io pregato de dovere scrivere alcune cose de abaco nece- 
sarie a' mercatanti, da tale che i preghi suoi me sono comman- 
damenti, non conmo presuntuoso ma per ubidire me sforcarò con 
Paiutorio de Dio, in parte sactisfare l'animo suo, cioè scrivendo 
alcune raigioni mercantesche conmo baracti, merriti e compagnie; 
cominciando a la regula de le tre cose seguendo positioni et, se 
a Dio piacerà, alcune cose de algebra l . 



The stemma on folio Ir identifies its commissioner as a 
member of the Picbi family of Borgo Sansepolcro 2 , a merchant 
no doubt, but one interested in the discipline of mathematics 



1 Piero, Trattato d'abaco, ed. G. Arrighi (Pisa 1970), fol. 3r. 

2 See above, Chapter I, n. 49. 



^ 



22 Chapter Two 

for its own sake. Indeed Piero's exercises extend in complex- 
ity far beyond the needs of commerce. 

Unrecorded and unknown until the twentieth century, the 
manuscript of Piero's Trattato d'abaco was identified by Gi- 
rolamo Mancini in the Biblioteca Laurenziana in Florence 
around 1917 3 . Recognizing the same hand in the pen drawings 
as in those of Piero's Prospettiva pingendi (Ambrosiana, C 
307) and his Libellus de quinque corporibus regularibus (Vat. 
Urb. lat. 632), he also identified the script as Piero's by com- 
paring it to the autograph Parma manuscript of the Prospec- 
tiva pingendi (Ms. 1576). Piero's Abaco manuscript derived 
from the collection of Lord Ashburnham in London, much of 
which had been bought from the Italian manuscript collector 
and mathematical historian, Guglielmo Libri. Thus Mancini 
suspected that Piero's manuscript once formed part of the Li- 
bri collection. Unbeknownst to Mancini, the proof of his well- 
founded suspicion lies in Libri's Histoire de la mathématique 
en Italie 4 . 

In the third volume of his Histoire, published in 1840, 
Libri transcribed a large part of the algebra section of Piero's 
Trattato d'abaco (folios 62r-79v) 5 , describing it as an anony- 
mous work. « On trouvera dans cette note des extraits de deux 
anciens manuscrits d'algebre qui m'appartiennent, et qui prou- 
vent que depuis long-temps les géomètres avaient tenté de ré- 
soudre les équations des degrés supérieurs au second » 6 . Libri 



3 G. Vasari, Vite cinque annotate, ed. G. Mancini (Florence 1917), 
pp. 210-14. 

4 Histoire des sciences mathématiques en Italie depuis la Renaissance 
des lettres jusqu'à la fin du dix-septième siede, 4 vols. (Paris 1838-41). For 
Libri's somewhat unorthodox methods of manuscript collecting, see L. Des- 
lisle, Les manuscrits des fonds Libri et Barrois à la Bibliothèque Natio- 
naie. Extrait du catalogue des ces manuscrits (Paris 1888). See also below, 
Appendix I, n. 3. 

5 Histoire, III (1840), 302-49. 

6 Ibid., p. 302. 



The « Trattato d'abaco » 23 

dateci the first manuscript, which was in fact Piero's, to the 
fourteenth century and located its origin in Tuscany. His dat- 
ing is incorrect, for the first manuscript was the Ashburnham 
treatise, not an earlier version of the work. One may be certain 
of this on several counts. The marginai notes in Ashburnham 
280, written in a sixteenth-century script, are reprinted in Li- 
bri's Histoire 1 . The penultimate problem in Libri's excerpt 
leads into Piero's geometrical section on regular bodies, in the 
Abaco, beginning on the next folio (80r). And finally, the same 
exercise recurs, in a Latin translation, in Piero's Libellus 
de quinque corporibus regularibus 8 . 

Libri's publication of Piero's work not only establishes the 
provenance of the manuscript. It also enlightens us concerning 
one branch of mathematics that occupied Piero, and it leads 
us to his medieval source. Libri considered these chapters in 
Piero's treatise important because they substantiated his belief 
that Italian mathematicians, since Leonardo Pisano, called Fi- 
bonacci, in the thirteenth century, had been attempting to solve 
algebraic equations higher than the second degree. 

The solution of the cubie equation constituted the first for- 
ward step in mathematics since the inventions of the Arabs, 
and it took place in Florence in the first half of the sixteenth 
century. The discovery of the solution was probably due to 
Niccolò Tartaglia. It was then published by Gerolamo Carda- 
no 9 . In any event, many mathematicians of the fifteenth and 



7 See Ministero della Pubblica Istruzione, Indici e cataloghi, Vili: 
I codici Ashburnhamiani della Biblioteca Medicea Laurenziana di Firenze, 
I, fase. 6, ed. E. Rostagno and T. Lodi (Rome 1948), 462, where the mar- 
ginalia are identified, « di mano del sec. XVI. » Libri reprints in italics the 
marginai note on fol. 64r of the Abaco (p. 309). Moreover, the additional 
solution found in the margin of fol. 76v of the Abaco is reproduced by 
Libri (p. 340). 

8 L'opera « De corporibus regularibus, » ed. G. Mancini (Rome 1916), 
Part I, no. 4 (pp. 491-92). 

9 A concise explanation of the cubie equation and its historical im- 



24 Chapter Two 

sixteenth centuries worked simultaneously to solve equations 
of the third, fourth, and fifth degrees. The ultimate solution 
of the cubie equation opened the way to modem algebra and 
modem algebraic equations. Piero's work in the Trattato d'aba- 
co must be seen as a contribution to this process of discovery, 
and thus Libri's publication illuminates the breadth, novelty, 
and experimental character of Piero's mathematics. 

Algebraic studies were first introduced into Italy by Leo- 
nardo Pisano, or Fibonacci, who codified the body of rules he 
learned in North Africa in his Liber abaci of 1202 10 . This 
work remained the fundamental mathematical text until the 
sixteenth century. Not only did abbaco writers refer to Fibo- 
nacci time and again, but his work provided the source for 
Piero 's investigations . 

Book XV of Fibonacci's Liber abbaci treats rules in geo- 
metry pertaining to algebra. The section most significant for 
Piero, entitled « De solutione quarumdam questionum secun- 
dum modum algebre et almuchabale, scilicet ad proportionem 
et restaurationem » ", instruets in the six forms of algebraic 
equations. Of these, Fibonacci writes, three are simple and 
three composite. The three simple equations are described 
thus: 

Primus quidem modus est, quando quadratus, qui census dicitur, 
equatur radicibus. Secundus quando census equatur numero; ter- 
tius quando radix equatur numero 12 . 



portance is given in E. Colerus, Piccola storia della matematica (Turin 
1962), pp. 153-58; also: G. Libri, Histoire, III, 148-81; L. Olschki, 
Geschichte der neusprachlichen wissenschaftlichen Literatur, III (1927; 
rpt. Vaduz 1965), 87-101; R. Taton, ed., The Beginnings of Modem Science 
from 1450 to 1800 (London 1964), pp. 35-42; H. G. Zeuthen, Geschichte 
der Mathematik im 16. und 17. Jahrhundert (1903; facsimile rpt. New 
York 1966), pp. 81-93. 

10 L. Pisano, Liber abbaci, voi. I of Scritti di Leonardo Pisano, ma- 
tematico del secolo decimoterzo, ed. B. Boncompagni (Rome 1857). 

11 Pisano, Liber abbaci, pp. 406 ff. 

12 Ibid., pp. 406-7. 



The « Trattato d'abaco » 25 

That is, in the first case, the square of the unknown equals 
a square root; in the second, the square of the unknown equals 
a whole number; in the third, the square root of the unknown 
equals a whole number. Then Fibonacci describes the three 
composite equations. 

Primus enim modus est, quando census et radices equantur nume- 
ro. Secondus, quando radices et numerus equantur censui, ter- 
cius modus est, quando census et numerus equantur radicibus 13 . 



In other words, in the first composite equation, the 
unknown squared plus a square root equal a whole number; 
in the second, the square root plus a whole number equal an 
unknown squared; and in the third, the unknown squared 
plus a whole number equal a square root. 

Piero begins his algebraic chapters (fol. 24r) in a similar 
manner. He defines square roots (« radici »), and then contin- 
ues to describe the simple and composite forms of equations. 

E, de queste, fa algibra 6 regule, tre semplici et tre composte. 
Le tre semplici sono quando nelle quistioni arismetrice o geume- 
trice se trova la cosa o vero radici equale al numero, o vero i 
censi equali a le cose, o vero il censo equale al numero. Però, 
quando le cose sono equali al numero, se dèi partire il numero 
per le cose e quello che ne vene vale la cosa. Et quando i censi 
sono equali a le cose, se dèi partire le cose per li censi e quello 
che ne vene vale la cosa. Et quando i censi sono equali al numero, 
se dèi partire il numero per li censi et la radici de quello che ne 
vene vale la cosa M . 



Composite equations are defined in the following way: 



13 Ibid., p. 407. 

14 Piero, Trattato d'abaco, fol. 24r. 



26 Chapter Two 

Et i composti sono quando i censi e le cose sono equali a li numeri, 
et quando i censi e i numeri sono equali a le cose, et quando 
il censo equale a le cose e al numero ... 15 . 



To Fibonacci's six forms, Piero adds an additional fifty- 
five possibilities that employ cubie equations, biquadratic equa- 
tions, and equations of the fifth degree 16 . Nearly half of 
these fifty-five rules are fully illustrateci with problems, and 
these are precisely the pages that Libri transcribed in his Hi- 
stoire de la mathématique . One example of Piero's more com- 
plex algebraic equations shows how he elabora'tes the six basic 
types. 

Quando le cose e i censi et i cubi e censi de censi sono equali 
al numero, se dèi partire nei censi di censi e poi partire le cose 
per il cubi et, quello che ne vene, recare a radici et ponere sopra 
del numero; et la radici de la radici di quella somma meno la 
radici de le cose che ne venne partite per li cubi, tanto vale la 
cosa 17 . 



Here we meet with a biquadratic equation — one includ- 
ing a simple unknown, the square of an unknown, the cube 
of an unknown, and an unknown raised to the fourth power. 
The problem that illustrates this rule concerns the interest rate 
to be paid on 100 Libre over four years. 

Just over a half a century after Libri's volume appeared, 
Piero's problems, as Libri had printed them, were closely ana- 



15 Ibid. 

16 Ibid., fols. 30r-78v. For an explanation of Piero's equations see S. 
A. Jayawardene, «'The Trattato d'abaco' of Piero della Francesca, » in Cul- 
tural Aspects of the ltalian Renaissance, Essays in Honour of Paul Oskar Kri- 
steller, ed. C. Clough (Manchester 1976), pp. 233-34, a work published 
after the present study was completed. 

17 Ibid., fol. 67v. 



The « Trattato d'abaco » 27 

lyzed by Moritz Cantor in his unsurpassed Vorlesungen uber 
Geschichte der Mathematik 18 . Even without the manuscript it- 
self, Cantor was so impressed with Libri's excerpts that he 
evaluated the work in the section of his book devoted to the 
development of mercantile mathematics. On the one hand, he 
points out some of Piero's false steps in attempting to solve 
cubie and other higher equations. But, on the other, Cantor 
maintained that, though in the last analysis unsuccessful, his 
anonymous algebraicist was an uncommonly good arithmeti- 
cian, indeed a very gifted mathematician, and he suspected, 
from the penultimate problem on triangles, that his man was 
equally talented in geometry 19 . Knowing only the single geo- 
metrical exercise Libri had published, Cantor speculated no 
further. It testifies to the novelty of Piero's work that Cantor 
devoted far more attention to Piero than to any other con- 
temporary algebraicist. 



2. The Contents of the geometrical sections of the « Abaco » 

We have seen that Piero's algebraic studies have their own 
significance in the history of mathematics. His geometry chap- 
ters, as Cantor suspected, reveal a no less gifted mind. More- 
over, Piero's geometry lessons furnish a key to understand- 
ing the entirety of his treatises. By carefully following the 
geometrical exercises in the Trattato d'abaco, by isolating cer- 
tain themes running through the entire work, and by determin- 
ing their sources, the interrela tedness of the Trattato d'abaco, 
the Libellus de quinque corporibus regularibus, and the Pro- 
spettiva pingendi may be seen. Thus a continuity exists in Pie- 
ro's written works, for ali three treatises stem from related 



18 2nd ed., II (Leipzig 1892), 144-50. 

19 Ibid., p. 149. 



28 Chapter Two 

interests. Piero's mathematics and his art need not be coun- 
ted as separate phenomena; they are components of the same 
intellectual enterprise, carried forward by a single individuai. 

Earlier we examined some of the geometrie themes which 
the normal trattato d'abbaco treated. The Fiorentine abbaco 
of Pier Maria Calandri, for instance, defines a solid simply 
as « quello che à lunghezza, larghezza et altezza », and it meas- 
ures objects like chests, wells, vats, columns, towers, trees, 
and walls 20 . A Neapolitan abbaco of circa 1478 divides ali sol- 
ids into two basic categories, columnar and pyramidal. Here, 
too, the illustrations are drawn from objects such as wells, 
barrels, and chests 21 . In contrast, the material which Piero del- 
la Francesca presents in the geometry chapters of his Trattato 
d'abaco distinguishes the work from its contemporaries. Piero 
treats almost exclusively the measurement, not of everyday ob- 
jects, but of abstract geometrical polygons and polyhedra. In 
his treatment of these two-dimensional figures and three-di- 
mensional solids, two centrai themes stand out — the golden 
section and the five regular bodies. The golden section exer- 
cises, like Piero's work in algebra, find their mathematical 
foundation in the work of Fibonacci, as well as in Euclid's 
Elements. The later exercises that deal with the five regular 
bodies are a direct consequence of Piero's Euclidean studies. 

The geometrical sections of Piero's Trattato d'abaco divide 
into two parts. The first (fols. 80r-104v) treats piane geometry 
— the measurement of two-dimensional polygons. The second 
(fols. 105r-120v) contains exercises in solid geometry — the 



20 P. M. Calandri, Tractato d'abbacho, ed. G. Arrighi (Pisa 1974), 
fols. 216v, 217r-223v. 

21 P. P. Muscarello, Algorismus: Trattato di aritmetica pratica e 
mercantile del secolo XV (Verona 1972), I, fols. 86v-87r: « Item corpi si 
sono di 2 manere principalminte, prima si è colonna, 2a è pirramide. Co- 
lonne sono corno casse, arche di grano, pozi, botte, tine. Pirramide sono 
corno tramoggia, monti di grano et omne figura ampia di sopto et apontata 
di sopra ». 



The « Trattato d'abaco » 29 

measurement of three dimensionai solids (often including 
three-dimensional bodies placed in a sphere). The rules for the 
measurement of triangles, squares, rectangles, parallelograms, 
pentagons, hexagons, octagons, and circles, taught on the first 
forty pages, establish the foundation needed to measure the 
succeeding solid geometrical bodies. Piero's definitions of the 
several polygons are remarkably clear, for instance, his defi- 
nitions of the pentagon and hexagon (fols. 88v-91v). 

El pentagono è una figura de 5 lati equali et de 5 anguli equali, 
de la quale figura i lati suoi s'à da la corda de l'angulo pentago- 
nico ... La figura de lo exagono è una figura de 6 lati equali a' 
semidiametro del circulo dove è descricta; e la superficie sua se 
devide in sei trianguli equilateri per li quali s'à la superficie sua 
mediante il catecto de uno de quelli trianguli. 

Equally clear are the problems and solutions Piero devis- 
es to explicate each of these piane geometrie figures. What 
is the area of the equilateral hexagon ABCDEF, Piero asks, 
on folio 91v, whose side is 6 braccia? By explaining the rela- 
tionship of the equilateral hexagon to the equilateral triangle 
and by referring back to his earlier lessons on the triangle, 
Piero makes the solution readily comprehensible. 

Fa'così. Tu sai che se devide in sei trianguli; pigia uno de questi 
trianguli che sono tucti equilateri et trova il suo catecto per 
quella via che te fu mostro nelli trianguli, che troverai che sirà 
la radici de 27. Et tu ài che a montiplicare il catecto per la metà 
de la basa ne vene la superficie del triangulo, adunqua pigia la 
metà de le base che sono 6, sirano 3, et è ciascuna 6 bracci, si- 
rano 18. Il quale 18 montiplica in sé farà 324, et questo mon- 
tiplica per lo catecto ch'è 27 fa 8748; et la radici de 8748 è la 
superficie del dicto exagono ABCDEF, che è 6 bracci per facia. 

Piero's precision and thoroughness in dealing with these 
abstract figures is striking when compared with the definitions 



30 Chapter Two 

of two-dimensional figures found in contemporary abbachi, 
where often the simplest definitions of very elementary forms 
sufficed. In the Neapolitan abbaco of 1478, fol. 86v, we read, 
« Item pentangoli che anno 5 lati e 5 angulli, exangulli à 6 
anguli e lati, multilateri anno multilati e multi anguli ». And 
in Pier Maria Calandri's contemporary Fiorentine Tractato, 
triangles, rectangles, and pentagons are described in the fol- 
lowing manner: « Le figure di tre linee rètte si dichono trian- 
goli, le figure di quattro linee rette si chiamano quadrangoli, 
le figure di 5 lati si chiamano pentaghoni... » (fol. 209v). 

Folios 105r-120r of the Trattato d'abaco present exercises 
in measuring solid geometrie forms and in demonstrating their 
proportional relationship to each other and to the sphere. Pie- 
ro treats the five Platonic-Euclidean bodies and the sphere, as 
well as irregular pyramids and solids of eight and fourteen 
faces, similarly placed in a sphere. The exercises are far more 
complex than the simple polygons in Alberti's Elementi di 
pittura and they, too, go far beyond the usuai practical exerci- 
ses of measurement in contemporary abbaco handbooks. 



3. The Golden Section, the Five Regular Bodies, and Piero 's 
« Abaco » 

The golden section first occurs in the piane geometry exer- 
cises of Piero's Abaco in the discussion of the pentagon; it 
recurs in the solid geometry section when Piero demonstrates 
the measurement of a dodecahedron (composed of twelve pen- 
tagons) in a sphere. Despite many specialized studies, there 
exists no general agreement as to how widespread the use of 
this proportion was among Renaissance artists 22 . Nonetheless, 



22 P. H. Scholfield, The Theory of Proportion in Architecture (Cam- 
bridge 1958), pp. 52-54, and bibliography, pp. 148-50; R. Wittkower, 



The « Trattato d'abaco » 31 

Piero's use of it speaks for its knowledge in artistic circles, 
and the clarity of Piero's exposition must have fostered its 
further diffusion. 

The peculiarity of the golden section is that, unlike most 
Renaissance proportions, it is incommensurable, that is geo- 
metrical, and cannot be expressed in terms of rational integers, 
that is of discrete whole numbers. To divide a line according 
to the golden section, or in its mean and extreme ratio, 
signifies that the resulting smaller section is in the same pro- 
portion to the larger section as the larger section is to the whole 
line. This can be reduced to a formula. Given the line ABC, 
where AB is the smaller section and BC the larger, AB : BC 
= BC : AC, or BC 2 = AB.AC. The proportion first appears on 
folio 88v, where the problem is to determine the side of a 
pentagon when the chord is 12 braccia. « Tu dèi sapere », 
Piero writes on folio 88v, «che 12 se dèi devidere secondo 
la proportione avente il meqqo et doi stremi, e la magiore par- 
te è il lato del pentagono esendo 12 la corda de Vangulo pen- 
tagonico ». In the result, the side of th e pentagon (equal to 
the larger section of the chord) is V 180-6. This type of appli- 
cation of algebra to geometrical principles (the process where- 
by Piero resolves the question) finds its roots in Fibonacci, 
whose work proved so decisive for Piero's algebra. Moreover, 
the division of the chord of a pentagon that equals 12 into its 
mean and extreme ratio is also found in Fibonacci. 

Book VI of his Practica geometriae (1220), entitled « In 
dimensione corporum », treats the division of a line into its 
mean and extreme ratio 23 . The first of Fibonacci's two 



« The Changing Concept of Proportion, » Daedalus (Winter 1960), p. 205- 
09. Cf. G. De Angelis d'Ossat, « Enunciati euclidei e 'divina proportio- 
ne' nell'architettura del primo Rinascimento, » in II mondo antico nel Ri- 
nascimento, Atti del V Convegno internazionale di studi sul Rinascimento, 
Firenze 1956 (Florence 1958), pp. 253-63. 

23 L. Pisano, Practica geometriae, voi. II of Scritti di Leonardo Pi- 
sano, matematico del secolo decimoterzo, ed. B. Boncompagni (Rome 1862), 



32 Chapter Two 

demonstrations corresponds to Piero's, for he divides the chord 
of a pentagon that measures 12 into its mean and extreme 24 . 
Fibonacci earlier in Book VI had made explicit the relation- 
ship between the division of a line according to the golden sec- 
tion and the pentagon: « In circulo de scripto penthagono equi- 
latero duorum eius angulorum corde sese invicem secent, utra- 
que secari necesse est. Ad idem punctum proportione medi] et 
extremorum maior utriusque pars equalis lateri penthagonico 
in omni circulo, cuius dyametrum rationali penthagoni equila- 
teri latus inrationale est, cui nomen linea minor » 25 . 

Book Vili of Fibonacci's Practica geometriae, « De quibus : 
dam subtilitatibus geometricis », is devoted to problems of the 
pentagon and the closely related decagon 26 . Fibonacci demon- 
strates, for example, how to find the length of a side of a pen- 
tagon inscribed in a circle whose diameter is 10, the diameter 
of a circle inscribed within a pentagon whose side is 10, and 
the side of a pentagon whose area is 50 27 . Similarly, Piero 
shows how to find the sides and area of a pentagon inscribed 
in a circle whose diameter is 12 28 . 

Other relations to Fibonacci can be discovered in Book 
XV of Fibonacci's Liber abbaci, a work devoted to the rules of 
geometry that pertain to algebraic questions *. Here Fibonac- 
ci demonstrated at length the division of numbers into two 
parts, often working wi'th the number 10 30 . One exercise di- 



p. 196: « Modus dividendi lineam media et extrema proportione. » 

24 Pisano, Practica geometriae, pp. 196-97: « Nam modus dividendi 
corda .be. media et extrema proportione est ut super quadratum eius, quod 
est .12. » 

25 Practica geometriae, pp. 161-62. 

26 Ibid., pp. 207 ff. 

27 Ibid., pp. 207-16. 

28 Piero, Trattato d'abaco, fol. 90r-v. 

29 Pisano, Liber abbaci, pp. 387-459, esp. 387: « Incipit capitulum 
quintum decimum de regulis geometrie pertinentibus, et de questionibus 
aliebre et almuchabale. » 

30 Liber abbaci, pp. 410-59. 



The « Trattato d'abaco » 33 

vides 10 into two parts, so that the first part divided by the 
second, plus the second divided by the first, and the whole 
multiplied by the first, yields 34 31 . It is noteworthy that Fibo- 
nacci's work includes three methods of resolving the division 
of 10 according to the golden section, so tha t the smaller section 
will equal 15 — v 125, and the larger ^125 — 5, and, he 
writes, there are stili other ways to perform this operation. 
« Possemus etiam in bis aliis moàis procedere; sed ista que 
diximus, suf fidanti et scis, secundum hanc divisionem, 10 di- 
visa esse media et extrema proportione; quia est sicut 10 ad 
maiorem partem, ita maior pars ad minorem » n . Subsequently, 
Fibonacci demonstrates that the resulting sections, 15 — 
Vi25 and ^125 — 5, multiplied according to the golden sec- 
tion formula, yield 10, the length of the line to be divided. 
Thus the methods of division were painstakingly established 
for other readers to study and to acquire skills in performing. 
Piero's exposition of the golden section is more exhaustive 
and it is, at least in part, based on Book XIII of Euclid's 
Elements. On folio 90r-v Piero demonstrates how to deter- 
mine the length of the sides and area of a pentagon inscribed in 
a circle whose diameter is 12. According to Euclid, XIII, 8, 
writes Piero, the side of a hexagon joined to the side of a dec- 
agon inscribed in the same circle results in a line divided into 
its mean and extreme ratio; furthermore, according to 
Euclid XIII, 9, the side of a hexagon joined to the side of a 
decagon is equal to the side of a pentagon inscribed in the same 
circle; and, following Euclid, XIII, 10, in the chord of a pen- 
tagon divided into its mean and extreme, the larger section is 
equal to the side of a pentagon. He also knew from Euclid 
that the side of a hexagon equals half the diameter of the circle 



31 Ibid., p. 419. 

32 Ibid., p. 438. Cfr. also Pacioli's exercises with the division of 10 
into its mean and extreme in the Divina proportione (Venice 1509), Part 
I, Chapters 10-22. 



34 Chapter Two 

in which it is inscribed. With the diameter of the circle as a 
known quantity, Piero could then proceed, basing his steps 
on the three propositions he had listed from Book XIII of 
Euclids's Elements. Applying algebraic methods to the geomet- 
rical principles of Euclid, Piero then arrived at the side of 
a pentagon by finding the side of a decagon inscribed in the 
same circle and adding to it the known side of a hexagon. And, 
he provides the formula for the golden section, « però che sia 
una linea cusì devisa che la menore parte montiplicata in tucta 
la linea faccia tanto quanto la maggiore parte montiplicata in 
se m e de s sima » 33 . 

A third example of the golden section in Piero's Abaco is 
elucidated in his discussion of the dodecahedron inscribed in 
a sphere 34 . This, of course, derives from problems concerning 
the pentagon, for the dodecahedron is composed of twelve 
regular pentagonal faces. Piero attempts to find one side of 
the face of a dodecahedron inscribed in a sphere whose axis 
is V48. 

Tu dèi sapere che il lato del cubo descricto in una medessima 
spera deviso, secondo la proportione avente mecco e doi stremi, 
che la magiore parte è il lato del corpo de 12 baxe pentagonali 
descricto in una spera. Et ài, per la 13a del 13 de Euclide, che 
la posanca del diametro de la spera è tripla a la posanga del lato 
del cubo da quella contenuto 35 . 



Piero then proceeds to find the side of the cube, which, 
according to Euclid 's formula, is one third the axis, or 4. The 
methods for dividing 10 into its mean and extreme had been 
taught, as we have seen, by Fibonacci in his Liber abbaci*. 



33 Piero, Trattato d'abaco, fol. 90v. 

34 Ibid., fol. HOr. 

35 Ibid. 

36 Pisano, Liber abbaci, pp. 410-59. 



The « Trattato d'abaco » 35 

The methods then existed, and Piero only had to apply them 
to the division of 4 in order to determine the pentagonal side 
of a dodecahedron. 

It is apparent that Piero was at home in Fibonacci's Liber 
abbati and Practica geometriae, works which constituted the 
principal sources for late medieval and early Renaissance math- 
ematics. The proportion that modem times calls « golden » 
and the Renaissance called « divine », Piero and Fibonacci 
leave without a name. Both refer to it, in Euclidean terms, 
more simply as the proportion for dividing a line in its mean and 
extreme ratio. Known since classical times, it was necessary to 
Euclid's construction of the pentagon and, consequently, also 
the dodecahedron. Fibonacci, and Piero in his wake, treat the 
proportion as one way of dividing a line, one way of several, 
and do not attribute the extraordinary characteristics to it that 
Luca Pacioli later does 37 . Nonetheless, Piero's students, as 
Fibonacci's before, whether they were merchants, mathemati- 
cians, architects, or artists, learned, if they followed their les- 
sons diligently, the division of a line in its mean and extreme 
ratio, that is the golden section, which later excited such in- 
terest and acquired the connotations of divine. 

Even though Piero does not explicitly cite Fibonacci in 
his Abaco, there is no doubt that Fibonacci provided one of 
the foundation stones of Piero's treatise. Piero's immediate 
sources are not to be found in earlier abbaco texts, where sol- 
id geometrical principles were generally applied to everyday 
objects, not to abstract geometrical forms. It is precisely these 
abstract geometrical figures and solids that most concern Pie- 
ro, well-versed as he was in Euclid's Elements. The applica- 
tion of algebraic methods to geometrical forms can be traced, 
in part, back to Fibonacci. Though Piero's relationship to Fi- 



37 Divina proportione, Part I, Chapters 10-22. 



36 Chapter Two 

bonacci has not been previously examined, the connection is, 
for a number of reasons, a logicai one to make. In a very ele- 
mentary sense Fibonacci's work was fundamental for the en- 
tire genre of abbaco textbooks. Contemporary abbaco writers 
frequently cite Fibonacci, referring to him as Leonardo Pisano, 
or, more familiarly, as « L. P. » Not only is Fibonacci's author- 
ity invoked, but a fifteenth-century treatise in applied arith- 
metic, « tratto da libri di Leonardo pisano », credi ted Fibo- 
nacci with the revival of mathematics in Tuscany. Its author 
reports that by around 1348, not so long after Fibonacci lived, 
there were enough flourishing scuole d'abaco in Florence to 
accommodate tens of hundreds of fanciulli 38 . 

The golden section is closely allied to the second theme 
that distinguishes Piero 's Trattato d'abaco, the five regular 
bodies. They are the only five equal-faced, equiangular solids 
that can be comprehended in a sphere, the tetrahedron, the 
cube, the octahedron, the icosahedron, and the dodecahedron. 
The tetrahedron is a pyramid composed of four equal, equilat- 
eral, and hence equiangular faces. The cube has six square 
faces. The octahedron, a doublé pyramid on a square base, 
has eight equal and equilateral triangular faces. The last two 



38 See the description of the scuole dell'abbaco in a treatise on practi- 
cal arithmetic (« Trattato di praticha darismetica tratto da libri di lionardo 
pisano, » Siena, Bibl. Pubblica Comunale, Cod. L. IV. 21, fols. 127r-v) 
of 1460, in B. Boncompagni, Intorno ad alcune opere di Leonardo Pisano 
(Rome 1854), p. 251: « Al fatto del ebaso di L. p. Dicho che L. p. fu 
huomo sottilissimo in tutte dispute, et secondo che si truoua lui fu il pri- 
mo, che ridusse allume questa praticha in toschana, che allora sandava per 
vie molte estrane, nientedimeno dassai tenpo manzi attui in questa nostra 
città furono schuole dabacho, che circha al 1348. ò veduto Trattato che dice 
in firenze essere più di 10. centinaia di fanciulli alle schuole dellabacho, 
che pocho inanzi fu lionardo. E anchora chome si uede lonsegnare loro era 
a modo antichi et quasi al modo che oseruono di presente e vinitiani, che .è. 
maraviglia al sufficienti (sic) maestri vi sono stati, et sono chome e non 
anno ridotto in una facile praticha tutto. » 



The « Trattato d'abaco » 37 

regular bodies approach more nearly the form of the sphere. 
The icosahedron has twenty equilateral triangular faces, and 
though the dodecahedron has only twelve faces, they take the 
form of regular pentagons. The perfection of these forms exer- 
cises an inevitable fascination, and that there are invariably, 
forever and ever, five and only five such bodies, is one of those 
eternai truths that have recurrently struck men as marvellous. 
Plato singled out these exclusive, essential forms as his sym- 
bols for the four elements and the universe 39 . And Euclid made 
them the culmination of his compendium of solid geometry 
in Book XIII of the Elements. Moreover, the regular bodies 
form the subject of two further books of the Elements, XIV 
and XV, which, though now excluded from the Euclidean 
canon, were considered as an integrai part of his work in the 
early Renaissance. Piero's Abaco exercises concerning the five 
regular bodies furnish the core of his later Libellus de quinque 
corporibus regularibus and bring us closer to his second basic 
source, Euclid's Elements, Books XIII-XV. (Here we must fol- 
low early Renaissance usage regarding Euclid's oeuvre.) 

The proportion that divides a line in its mean and extreme 
ratio, the « golden section », is fundamental to Book XIII of 
Euclid's Elements, on the regular bodies, for it is the subject of 
the first six propositions, and it is necessary to the construction 
of the dodecahedron. Luca Pacioli made the connection be- 
tween the regular bodies and the golden section explicit in the 
title to Book XIII of his 1509 edition of Euclid's Opera, « Li- 
ber tertiusdecimus Euclidis de admiranda vi linee secundum 
proportionem haventem medium duoque extrema divise et 
quinque corporum regularium formatione » m . With Piero the 



39 See above, Chapter I, n. 52. 

40 Euclidis Megarensis philosophi acutissimi ... Opera, ed. L. Pacioli 
(Venice 1509), fol. 122r. Pacioli had also taught the proportional measure- 
ment of the regular bodies in Chapters 26-31 of the Divina proportione. 



38 Chapter Two 

connection is left implicit, as, for instance, in his discussion 
of the pentagon and dodecahedron, which we examined above. 
Nonetheless, understanding regular bodies required under- 
standing the division of lines into their mean and extreme 
ratios. 

The second part of the geometry section of Piero's Abaco 
(fols. 105r-120r) treats regular and irregular bodies, their re- 
lationship to a sphere and to one another. In this section Pie- 
ro's relationship to Euclid becomes evident in such references 
in his text as « conmo per la penultima del primo de Euclide 
se demostra ». Similar references, however, abound in Fibo- 
nacci, and we may not exclude a priori the possibility of an 
indirect transmission of Euclidean geometry via Fibonacci. 
Nonetheless, Piero's first-hand knowledge of the works of 
Euclid can be demonstrated by a dose reading of his text. 

Piero begins his solid geometry section, « Il corpo à tre 
demensioni, cioè longhegga, larghegga e profondità; et sono 
de più ragioni, benché io non intenda dire se none de alcuni 
conmo è de' 4-base triangulare et de' cubi e de' tondi speri- 
ci » \ Though Piero does not intend to discuss ali solids, those 
with equal sides and faces will be included, and these are the 
five regular bodies. Piero first discusses the cube and the tet- 
rahedron, including their interrelationships, their relations to 
the sphere, and some of their irregular derivatives. Only when 
he reaches the octahedron does he reveal his intention to dis- 
cuss ali five Platonic-Euclidean solids. « De questo corpo», 
he writes, « facilemante s'àno le sue mesure, però non de' cre- 
decti dire niente et è remaso de rieto per questo; ma, perché 
ci sieno tucti 5 li corpi regulari non voglio lassare » 4 \ In the 
remaining exercises in solid geometry Piero redeems his prom- 



41 Piero, Trattato d'abaco, fol. 105r. 

42 Ibid., fol. 114r. 



The « Trattato d'abaco » 39 

ise. AH five regular bodies are explained, the tetrahedron 
(fol. 105r-v), the cube (fol. 106r-v), the dodecahedron (fols. 
HOr-lllv), the icosahedron (fols. 112r-113v), and the octa- 
hedron (fol. 114r). 

Fibonacci 's discussion of the regular bodies in his V radica 
geometriae is neither so extensive as Piero's nor as complete. 
Having listed only three of the five solids, Fibonacci directs 
his reader to Euclid, who teaches their construction in a sphere. 
« Solida multarum basium sunt multimoda, ex quibus sunt 
solida Vili, basium, et XII. basium, et XX. basium equalium, 
que Euclide s in xiiii libro infra speram construere docet » 43 . 
Save for a short excursus on the dodecahedron and icosahe- 
dron, Fibonacci is more concerned with the measurement of 
practical bodies 44 . 

For an extensive explanation of the regular bodies we are 
obliged to take Fibonacci's advice and turn to Euclid. There 
we find the evidence (especially in light of Piero's later trea- 
tise devoted to the regular bodies) that Piero, too, went back 
to the ancient source, for the theorems that He at the heart 
of Piero's work are laid out consecutively in propositions 13 
to 18 of the thirteenth book of Euclid's Elements. 

Book XIII of Euclid's compendium instructs in the divi- 



43 Pisano, Practica geometriae, p. 159. 

44 Fibonacci writes (p. 158) that bodies are of many kinds such as 
solids, cones, pyramids, columns, and spheres: « Corporum quiàem genera 
sunt plura, ex quibus sunt hec: Solidi, Seratilia, Piramides, Columne, Spe- 
re, et eorum partes, nec non et corpora, que multorum basium nuncupantur, 
et describuntur circa speras: proprie quidem solidum est, quod latum lon- 
gum et altum babet; et constat ex sex superficiebus: ut sunt t assilla: scri- 
nea, cisterne et similia. » In the solid geometry section of the Liber abbaci 
(pp. 403-05), Fibonacci deals with the problems of water displaced in a 
cistern into which pyramidal-shaped stones, columns, or spheres are placed 
Furthermore in finding the area of a square based pyramid the problem is 
couched in the familiar, applied mathematical terms of three men painting 
the pyramidal roof of a palace (p. 405). 



40 Chapter Two 

sion of a line into the golden section (propositions 1-6). It also 
treats the construction of the regular bodies in a sphere, togeth- 
er with their proportional relationship to the sphere (propo- 
sitions 13-17) and to one another (proposition 18). These 
propositions relate directly to the regular bodies discussed by 
Piero. Proposition 14, for instance, is required to solve Piero's 
problem with the octahedron (fol. 114r), « find the side of an 
octahedron comprehended in a sphere whose diameter is 10 ». 
Euclid's proposition establishes that the square on the diame- 
ter of the sphere (100) is doublé the square on the side of 
the octahedron (or 2x 2 ). Hence 100 = 2x 2 , or the side of the 
octahedron equals ^50. In addition, Euclid's proposition 15 
is necessary to the solution of Piero's problem determining 
the side of a cube comprehended in a sphere whose diameter 
is 7 (fol. 107r). Knowing from Euclid that the square of the 
diameter (or 49) equals three times the square of the side of 
a cube (3x 2 ), Piero quickly demonstrates the answer. 

For one more instance, Proposition 9 of Book XIV of 
the Elements is fundamental to Piero's finding the area of a 
dodecahedron whose side is 4 (fol. HOv), Piero writes: « Tu 
ài per la 9 a del 14° de Euclide, che li tre quarti del diametro 
del circulo dove è descricto il pentagono, montiplicato in cin- 
que sexti de la linea che soctotende Vangulo pentagonico prova 
che fa la superficie del pentagono ». There are many other 
such examples in Piero's text which relate directly to a manu- 
script source of Euclid's Elements. 

In the problems concerning the dodecahedron and icosa- 
hedron (fols. 110r-113v), Piero utilized extensively Book XIV 
of the Elements, whose propositions 5, 6, 7, 8, 9, and 10 illus- 
trate the relationship of these figures to each other as well 
as to the sphere that comprehends them. 

The theme of the regular bodies reaches its culmination, 
of course, in the Libellus de quinque corporihus regularibus, 
and the following chapter will provide a detailed comparison 
of this manuscript with the Abaco. For the present the Tratta- 



The « Trattato d'abaco » 41 

to d'abaco may be defined as Piero's initial (or at least early) 
foundation for his Libellus, a work even more clearly Euclidean 
in character. 



4. The « Abaco » and the « Prospectiva pingendi » 

To ali appearances Piero's Trattato d'abaco is a work of 
pure mathematics. In it Piero never mentions the relevance of 
solid geometry to artists, yet it is precisely the measurement 
of the surfaces of geometrical bodies that is one of the primary 
concerns of the science of perspective. Of the three principal 
parts of painting, « disegno, commensuratio et colorare », Pie- 
ro limits himself to the second in his De prospectiva pingendi: 
« Commensuratio », he writes, « diciamo essere essi profili et 
contorni proportionalmente posti nei luoghi loro », and this 
is also called « prospectiva » 45 . He will not treat color, but 
instead, « tracteremo de quella parte che con line angoli et 
proportioni se pò dimostrare, dicendo de puncti, linee, super- 
fide et de corpi » 4é . The first of the three books on perspective 
treats « puncti, de linee et superficie piane », the second, 
« corpi chubi, de pilastri quadri, de colonne tonde et de più 
facce », and the third, « le teste et capitelli, base, torchi de più 
base et altri corpi diversamente posti » 47 . The pure mathemat- 
ical definitions given in an abbaco are then redefined for the 
artist. For the mathematician, Piero writes in the Prospectiva 
pingendi: 

Puncto è la cui parte non è, secondo i geumetri dicono essere 
inmaginativo; la linea dicono avere lunghezza senza latitudine. 



45 De prospectiva pingendi, ed. G. Nicco Fasola (Florence 1942), p. 63t 

46 Ibid., p. 64. 

47 Ibid., p. 65. 



42 Chapter Two 

And fot the artist, he writes: 

Dirò adunqua puncto essere una cosa tanto picholina quanto è 
posibile ad ochio comprendere; la line dico essere extensione 
da uno puncto ad un altro, la cui larghezza è de simile natura 
che è il puncto. Superficie dico essere larghezza et longhezza 
compresa da le linee. Le superficie sono demolte ragioni, quale 
triangola, quale quadrangola, quale tetragona, quale pentagona, 
quale exagona, quale octagona, et quale de più et diverse facce, 
commo per figure ve se dimostrare 48 . 



The introduction to Book II of the perspective treatise 
testifies most clearly to the relationship between the De pro- 
spettiva pingendi and the geometry of the Abaco. 

Corpo ha in sé tre dimensioni: longitudine, latitudine et altitu- 
dine; li termini suoi sono le superficie. I quali corpi sono de 
diverse forme, quale è corpo chubo, quale tetragono che non sono 
de equali lati, quale è tondo, quale laterato, quali piramide la- 
terate, et quale di molti et diversi lati, sicommo ne le cose na- 
turali et ancidentali se vede. De li quali in questo secondo [libro] 
intendo tractare de la loro degradationi, nelli termini posti da 
l'occhio socto angoli compresi, facendo de alcune superficie de- 
gradate nel primo lor base 49 . 



These are the solids, the cubes, pyramids, and spheres, 
the many-faced bodies that Piero had described earlier. For 
them he had invented mathematical exercises in measurement 
in the Ashburnham Abaco. Finally, the importance of these geo- 
metrical bodies for painting is attested to in the introduction 
to Book III of the Prospettiva pingendi: « Et perché la pittu- 
ra non è se non dimostrationi de superficie et de corpi degrada- 



« Ibid., pp. 65-66. 
49 Ibid.. p. 100. 



The « Trattalo d'abaco » 43 

// o acresciuti nel termine, posti secondo che le cose vere ve- 
dute da l'occhio socio diversi angoli s 'apresentano nel dicto 
termine... » 50 . 

The mathematical rules for the correct, scientific measure- 
ment of regular and irregular geometrical solids demonstrated 
in the Trattato d'abaco were, in a way, preliminary and nec- 
essary to a thorough understanding of the Prospectiva pin- 
gendi. As Piero wrote, a painter needed to know the signifi- 
cance of lines and angles in order to under stand perspective. 
Hence the study of regular solids, and their measurement and 
placement in a composition, was of interest to mathematicians 
and painters alike. The theorems that He at the heart of Book 
XIII of Euclid's Elements instruct the mathematician in the 
relationship of one regular body to another and train the paint- 
er to see and reproduce the inanimate bodies found in nature. 
A few years later, Luca Pacioli's Divina proportione (in es- 
sence a treatise based on the Euclidean-Platonic five regular 
bodies) was dedicated equally to students of «philosophia, pro- 
spectiva, pictura, sculptura, architectura, musica, e altre ma- 
thematice » 51 . 



50 Ibid., pp. 128-29. 

51 Divina proportione, title page. 



III. 

The «Libellus de quinque corporibus regularibus»: Regular 
and Irregular Bodies 



The Libellus de quinque corporibus regularibus (Vat. Urb. 
lat. 632), dedicated to Guidobaldo da Montefeltro, Duke of 
Urbino, was composed in the decade between 1482 and Pie- 
ro's death in 1492 l . The words of praise and appreciation 
which Piero lavishes on the family that so often sponsored 
him, in his dedication to Guidobaldo, are not entirely originai 
for they are based on a Vitruvian formula. Nonetheless, the 
sentiments he expresses are an accurate reflection of Piero's 
dose relations with the young duke and with his late father, 
Federigo da Montefeltro. Moreover, in his dedication to Gui- 
dobaldo, Piero specifies the primary innovation of his Libel- 
lus and indicates its connection to his earlier De prospectiva 
pingendi. 

In writing to Guidobaldo that the lasting fame of many 
artists rested primarily on the men they served, Piero para- 
phrases the introduction to Book III of Vitruvius' De architec- 
tura. 

Inter antiquos pictores et statuarios, Guido princeps insignis, 
Policretum, Phidiam, Mironem, Praxitelem, Apellem, Lisippum, 



1 L'opera « De corporibus regularibus, » ed. G. Mancini (Rome 1916). 
In 1482 Federigo da Montefeltro died, and his son Guidobaldo succeeded 
him as Duke of Urbino. 



The « Libellus de quinque corporibus regularibus » 45 

ceterosque qui nobilitateci ex arte sunt consecuti, non ob aliud 
digniores fuisse, et apud suos maiorem gratiam, apud vero poste- 
ritatem memoriam et famam diuturniorem, Aristomene, Thasio, 
Polide, Chione, Pharaxe, Boeda, ceterisque, qui non minori artis 
studio, ingenio, solertia, et industria fuerunt, habuisse perhibent, 
nisi quod ii aut civitatibus magnis, aut regibus, aut principibus, 
virtutis experimentatae opera fecerunt 2 . 



Then, continuing his praise of the house of Montefeltro, 
Piero reveals the incentives that led him to write and the pur- 
poses that lay behind his treatise. 

Cum autem opera picturaeque meae a splendidissimo et fulgen- 
tissimo sidere, et maiore nostri temporis luminare optimi genito- 
ris tui totum quicquid habent claritatis assumpserint: non ab re 
visum fuit opusculum quod in hoc ultimo aetatis meae calculo, 
ne ingenium inertia torpesceret, in mathematica de quinque cor- 
poribus regularibus edidi nummi tuo dedicare, ut et ipsum ex 
obscuritate sua a claritate tua illustretur. Nec dedignabitur cel- 
situdo tua ex hoc iam emerito, et fere vetustate consumpto agello, 
unde et illustrissimus genitor tuus, uberiores percepit, hos exiles 
et inanes fructus suscipere, et libellum ipsum inter innumera am- 
plissimae tuae, paternaeque bibliothecae volumina penes aliud 
nostrum de Prospectiva opusculum, quod superioribus annis edi- 
dimus, prò pedissequo et aliorum servulo, vel in angulo collocare. 
Non enim solent non admitti quandoque in opulentissima et 
lautissima mensa, agrestia, et a rudi et inepto colono poma su- 
scepta. Poterit namque, saltem sui novitate, non displicere. Ete- 
nim licet res apud Euclidem, et alios geometras nota sit, per 
ipsum tamen nuper ad arithmeticos translata est. Eritque pignus 
et monumentum mei in te, inclitamque prosapiam tuam antiqui 
amoris et perpetuae servitutis 3 . 



We learn here of Piero's advanced age and frail physical 



2 De corporibus regularibus, p. 488. Cf. Vitruvius, De architectura 
(III, Preface). 

3 De corporibus regularibus, p. 488. 



46 Chapter Three 

condition. And two other facts important £or understanding 
the Libellus emerge in the dedication. First, Piero wished to 
have the Libellus shelved together with the Prospectiva piti- 
gendi in the Urbino Library, and, second, Piero believed his 
achievement in the Libellus to be the application of arithme- 
tical principles to the geometrical theorems of Euclid. 

Piero's wish that his two treatises stand together indicates 
that he intended the study o£ the Libellus to be associated 
with that of the Prospectiva pingendi. And, indeed, the works 
are united by a similarity of structure and content. The rela- 
tionship of the measurement of regular and irregular bodies 
to the study of perspective was touched on in the preceding 
discussion of the Trattato d'abaco. However, it is the perspec- 
tive treatises of Piero's successors, with their demonstrations 
in constructing and rendering in perspective stereometrical 
solids that make this connection explicit. 

The application of arithmetical or algebraic exercises to Eu- 
clidean theorems, the novel aspect of Piero's treatise, is based 
on principles already worked out in his Abaco. But, while 
much of the Libellus derives from the Abaco, it represents a 
further development in Piero's mathematical studies, for a new 
consideration of Book XV of Euclid's Elements forms the ba- 
sis of one section of the treatise. 



1. Regular bodies: The « Libellus de quinque corporibus », 
Parts I-III 

Over half of the exercises in the Libellus, Parts I-III, had 
previously been demonstrated in the geometry section of Pie- 
ro's Abaco (fols. 80r-120r) 4 . In Part I of the Libellus — which 
treats two-dimensional figures, triangles, squares, pentagons, 
hexagons, octagons, and circles — some thirty-three of the 

4 Trattato d'abaco, ed. G. Arrighi (Pisa 1970). 



The « Libellus de quinque corporibus regularibus » 47 

fifty-five exercises can be found in the Trattato d'abaco (nos. 
3, 4, 8, 11, 14-25, 28-30, 34-40, 42-47, 49). Fewer than 
half of these, aroimd sixteen, have been reworked (nos. 3, 14, 
20, 24, 25, 28, 29, 35-41, 43, 44). Seventeen of the problems 
have been taken over much as they were (nos. 4, 8, 11, 15-19, 
21-23, 30, 34, 45-47, 49). For instance, Libellus, I, 15 reads: 

Eius quadrati cuius superficies est bis tanta quanta sunt latera 
sua quatuor, latus invenire? 

Habes in Algebra quod quadrata figura capitur prò censu, et 
suum latus habetur prò radice, id est prò re. Igitur die sic. Est 
census qui est 4 res, quia aequat duplum 4, quod est 8 res. Et 
regula dicit quod partiaris res per census; et res tantum valet quan- 
tum inde provenit. Partiare 8 [res] per 1 [census] eveniet 8; 
et 8 valebit res quae posita fuit prò latere quadranguli: igitur 
fuit 8. Multiplica in se 8 fient 64. Et sua quatuor latera, quorum 
quodlibet, 8, efficit 32, et quadratum est 64, qui est duplum ad 
32, qui sunt quatuor sua latera, quod propositum fuit 5 . 

This is clearly based on the Trattato d'abaco, folio 83r (2). 

Egl'è uno quadrato che la superficie sua è doi cotanto che i suoi 
4 lati. Domando quanto è il suo lato. 

Tu ài nell'argebra che il quadrato se intende il censo e il suo lato 
se intende radici cioè la cosa; adunque dirai che meco censo sia 
equale a 4 cose. Dunqua 1/2 censo è equale a 4 [cose]; reduci 
ad 1 censo, arai 1 [censo] equale ad 8 cose; parti le cose per 
li censi ne vene 8, et 8 vale la cosa. Noi dicemmo che uno lato 
era 1 [cosa] adunqua fu 8; montiplica 8 in sé fa 64 et i suoi 4 
lati, cioè 8 per lato, fa 32; adunqua il quadrato, che è 64, è doi 
cotanto che 32, che sono i 4 lati. 



5 The Libellus problems are here numbered as they appear in Mancini 's 
book, and in the Vatican 632 manuscript. I have numbered each of the 
exercises in Piero's Abaco in parentheses, according to the order of its ap- 

? earance on the individuai folio, following Arrighi's folio numeration. See 
able II for the correspondences between the problems in the Libellus and 
the Trattato d'abaco. 



48 Chapter Three 

Similarly, the foundation for the Libellus, I, 49, lies in 
the Trattato d'abaco, fol. 94r (1). Thus, in the Libellus we 
read: 

Si de diametro circuii, qui est ulnarum 10, linea ulnarum 9 1/2, 
resecant ulnas 3; in qua sui parte linea dividat quaerendum est. 
Sic age: Multiplica invicem diametri partes, quarum una est 
3, altera 7. Multiplica per 7, erit 21. Nunc si dicas: fac de 9 
1/2 partes duas huiusmodi, quarum ad invicem facta multipli- 
catio reddat 21. Ponamus partem unam esse 1 rem, et alteram 
9 1/2, dempta 1 re. Multiplicata semel una res in 9 1/2, deducta 
1 re, erunt res 9 1/2, dempto 1 censu. Et tu vis 21. Instaura 
partes: habebis 9 1/2 aequantem censum 1 et 21 numerum. Di- 
midiando, res erunt 4 3/4; multiplicatae in se, fient 22 9/16. 
Deduc numerum, qui est 21: remanet 1 9/16, cuius radix, de- 
ductis rebus dimidiatis, valet res, quae fuit de partibus lineae. 
Et altera fuit 4 3/4 addita radice ipsius 1 9/16. Itaque habes 
quod pars una est 4 3/4, remota radice ipsius 1 9/16. Et alia 4 
3/4 addita radice ipsius 1 9/16, id est una 3 1/2, alia 6. 



And, the corresponding passage in the Abaco, fol. 94r (1), 
is as follows: 

Egl'è uno tondo che il suo diametro è 10, et una linea che è 
9 1/2 il sega, cioè ne tagla 3 bracci none ad angulo recto. Do- 
mando in che parte è devisa la linea dal diametro. 
Fa' così. Montiplica le parti del diametro l'una con l'altra, cioè 

3 via 7 fa 21. Hora di' così: devidi 9 1/2 in tale parte che 
montiplicata l'una per l'altra facci 21. Mecti una parte essere 1 
cosa e l'altra 9 1/2 meno 1 cosa, hora montiplica 1 [cosa] via 
9 1/2 meno 1 [cosa] fa 9 1/2 [cosa] meno 1 censo, e tu voi 
21. Restora le parti; da' ad omni parte 1 censo, arai 1 [censo] 
et 21 equale ad 9 1/2 [cose]. Demecca le cose sirano 4 3/4, 
montiplicale fa 22 9/16, tranne il numero che è 21, resta 1 9/16. 
E la radici de 1 9/16 meno del dimecamento de le cose, che fu 

4 3/4, vale la cosa. Et fu dicto che una parte era 1 [cosa], 
dunqua fu 4 3/4 meno radici de 1 9/16, e l'altra parte fu 4 3/4 
più radici de 1 9/16. 



The « Libellus de quinque corporibus regularibus » 49 

Aside from these correspondences, the Vatican Libellus, 
Part I, contains new references to Euclid in exercises 14, 34, 
39, and 46, ones which do not appear in the corresponding 
problems in the Trattato d'abaco. 

Part II of the Libellus teaches the measurement of the 
regular bodies (the tetrahedron, cube, octahedron, dodecahe- 
dron, and icosahedron) contained in a sphere. Of the thirty- 
seven exercises in Part II of the Libellus, twenty-six can be 
located in the Trattato d'abaco (nos. 4-16, 18, 21, 26-36). 
Fourteen of these have been reworked to some degree (nos. 
4, 6, 11, 12, 14-16, 18, 21, 26-28, 30, 35). Twelve remain 
the same (nos. 5, 7-10, 13, 29, 31-34, 36). Several examples 
will illustrate these cases. 

Viginti basium corporis triangularium aequilaterarum cuius su- 
perficies est 200 brachiorum, quantum sit latus est perquirendum. 
In superiore dictum est quod si latus unius basis est brachiorum 
4, cathetus est radix 12. Et eius basis superficies erit radix 
48, sicut habes per Ilam primi. Nunc habes quod 20 basium est 
200 ulnarum: ideo partire 200 per 20, evenient 10, et 10 est 
superficies unius basis, quia proportio a superficie ad superficiem 
est dupla ad proportionem unius lateris unius superficiei, ad 
latus alterius superficiei quando sunt similia. Itaque die: si ra- 
dix 48 superficiei dat de latere 4, quantum dabit 10 de superficie? 
Reducas 4 ad radicem radicis, fiet 256. Et refer 10 ad radium 
fiet 100. Die itaque: si 48 de superficie dat de latere 256, quan- 
tum dabit 100 de superficie? Multiplica centies 256, fient 25600: 
partire per 48, evenient 533 1/3. Et radix radicis erit per quod- 
libet latus corporis 20 basium, cuius superficies est 200 ulnarum. 



This is Libellus, II, 34, and it is readily perceived that it is 
only a restatement of the Trattato d'abaco, fol. 113r (2), which 
follows. 

Egl'è uno 20-baxe triangulare equilatero, che la superficie sua 

è 200 bracci. Domando de' suoi lati. 

Tu ài che se il lato de una basa è 4 bracci, il catecto è radici 



50 Chapter Three 

de 12, e la superficie de quella baxa è radici de 48. E tu ài che 
20 braci è la superficie de' 20-baxe, però parti 200 per 20 ne 
vene 10 et 10 è superficie de una baxa. Et perché la proportione 
de la superficie a la superficie è doppia a la proportione del lato 
al lato de quelle superficie, però di': se radici de 48 de superficie 
dà de lato 4, che darà [10] de superficie? Reca 4 a radici de 
radici fa 256, et reca 10 a radici fa 100; montiplica 100 via 256 
fa 25600, parti per 48 ne vene 533 1/3. Et la radici de la ra- 
dici de 533 1/3 sirà per lato il 20-baxe triangulare, che la su- 
perficie sua è 200 bracci. 



The following quotation illustrates a different situation. 

Quando octobasis triangularis in quadratura esset 400, quaeritur 
de diametro sperae illum continentis. 

Age sic. Invenias sperata, cuius diameter notus sit, et hic sit 
7, qui per XXIIIIam huius dat de quadratura octobasis 57 
1/6. Reducas 7 ad radicem cubam, fiet 343. Dicas igitur sic. Si 
57 1/6 quadraturae dat de diametro 343, quantum dabit 400 de 
quadratura? Multiplica 343 cum 400, evenient 137200, quem 
partiare per 57 1/6, evenient 2400. Et radix cuba huius 2400 
est diameter sperae quae octobasim circumscribit, quadraturam 
habentem quantitatem 400. 



This is Libellus, II, 26, which has been completely re- 
worked from Trattato d'abaco, fol. 1 14v ( 1 ), which is given be- 
low, though the answer remains the same. 

Egl'è uno 8-baxe triangulare equilatero, che è quadrato 400 
bracci. Domando del diametro dela spera che il circumscrive. 
Tu ài, per la passata, che 1333 1/3 de quadratura dà de dia- 
metro 20; reca 20 a radici cuba fa 8000, però di': se 1333 1/3 
dà 8000, che darà 400? Montiplica 400 via 8000 fa 3200000, 
il quale parti per 1333 1/3; fa' tergi de le parti, arai 9600000 
[a] partire per 4000 che ne vene 2400. Et la radici cuba de 
2400 sirà il diametro de la spera che contene Pocto-baxe trian- 
gulare, che la sua quadratura è 400 bracci; ch'è il proposto. 



The « Libellus de quinque corporibus regularibus » 51 

In contrast, the first part of Part III (exercises 1 through 
13) reveals Piero breaking new ground, and it evidences his 
total immersion in Book XV of Euclid's Elements 6 . Problems 
14 through 29 comprise the exercises on the sphere to which 
Piero refers in his introduction. He prefaces this section with 
a paragraph in which he defines the sphere according to Euclid 
and Theodosius — a sphere is a round body, its greatest cir- 
cumference is determined by its axis, and its area is determined 
by its axis and circumference 7 . 

Nonetheless, of the sixteen problems Piero enumerates 
on the sphere (Part III, nos. 14-29), thirteen are already 
present in the Trattato d'abaco (nos. 15-17, 19-25, 27-29). 
Five of these problems are much the same in both treatises 
(nos. 17, 23-25, 28). We may compare, for instance, Libellus, 
III, 24, with the Trattato d'abaco, fol. 118v (1). No. 24 of 
the Libellus reads: 

Speram cuius axis est 14, linea plana 9 ulnarum dividat. Quo in 
loco resecet axem pervidendum est. 

Habes speram ABCD, cuius AD est axis, et linea BC scindit 
axem in puncto E. Et quia resecat ad angulum rectum divisa est 
linea BC per aequa in puncto E. Igitur BE est 4 1/2, quae est 
medietas BC, quae est 9. Multiplicato 4 1/2, in se, efficit 20 
1/4. Nunc si dicas de diametro sive axe, quae est 14, facias partes 
duas, quarum ad invicem multiplicando reddat 20 1/4. Ideo die 
quod una pars sit 1 res, altera sit 14, dempta 1 re. Multiplices 
1 rem cum 14, dempta 1 re, fient 14 res, remoto 1 censu: et tu 
vis 20 1/4. Restaura partes: habebis 1 censum, et 20 1/4 nu- 
merum aequantem 14 res. Dimidiando res, erunt 7, quae in se 
multiplicatae reddunt 49, de quo deme numerum, qui est 20 
1/4, supersunt 28 3/4. Et radicem 28 3/4, demptis rebus di- 



6 Euclidis Megarensis philosophi acutissimi ... Opera, ed. L. Pacioli 
(Venice 1509), fols. 141r-144v. 

7 De corporibus regularibus, p. 549: « Est igitur spera, auctore Euclide, 
transitus circumferentiae dimidii circuii, quotiens fixa diametro, quoasque 
ad locum suum redeat circumducitur. A Theodosio autem sic deffinitur: 
Spera est solidum corpus una superficie contentum, in cuius medio punctus 
est a quo omnes lineae ductae ad circumferentiam sunt aequales. » 



•52 Cbapter Tbree 

midiatis, valet res. Ergo pars una fuit 7, dempta radice 28 3/4. 
Altera pars 7, addita radice 28 3/4. Scindit igitur de axe 7, 
deducta radice 28 3/4. 



It is a simple matter to see in this problem the reflection 
of the corresponding problem in the Trattato d'abaco, fol. 
118v(l). 

Egl'è una spera che il suo diametro è 14 bracci; meno una linea 
piana, ch'è 9 bracci, segante l'assis de la spera ad angulo recto. 
Domando quanto taglarà de l'assis. 

Tu ài la spera ABCD et AD è diametro; et BC il sega in puncto 
E et, perché la sega ad angulo recto, è devisa BC per equali in 
puncto E; dunqua BE è 4 1/2, ch'è una parte de 9; che mon- 
tiplicato 4 1/2, ch'è una parte de 9; che montiplicato 4 1/2 in 
sé fa 20 1/4. Hora di' mo così: famme del diametro, ch'è 14, 
do' parti che montiplicata l'una per l'altra faccia 20 1/4. Dunqua 
mecti una parte 1 cosa e l'altra 14 meno 1 cosa; montiplica 1 
cosa via 14 meno 1 cosa fa 14 cose meno 1 censo, et tu voi 20 
1/4. Restora le parti; da' ad omni parte 1 censo, arai 14 cose 
equale ad 1 censo e 20 1/4 numero. Demegca le cose sirano 7, 
montiplicale in sé fa 49, tranne il numero ch'è 20 1/4, resta 
28 3/4. Et la radici de 28 3/4 meno del dimenamento de le cose, 
che fu 7, vale la cosa. Dunqua una parte fu 7 meno radici de 
28 3/4, l'altra fu 7 più radici de 28 3/4. 



Eight of the sixteen problems present in both texts have 
been reworked (nos. 15, 16, 19-22, 27, 29). Libellus, III, 19, 
for example, reads: 

Si de quadratura sperae, cuius axis est 7, fiat quadratura cubi, 
quantum sit eius cubi latus perquirendum est. 
Reducas speram in quadrum, cuius axis est 7. Et per XVIam 
huius habes quod eiusmodi sperae quadratum est 179 2/3. Igi- 
tur erit cubi latus radix cuba 179 2/3. Alia ratione potest de- 
monstrari, videlicet cum proportione. Ea enim proportio est 
inter latus cubi et diametrum sperae eiusdem quadraturae, quae 
est inter radicem cubam 343, et radicem cubam 179 2/3. Nam 



The « Libellus de quinque corporibus regularibus » 53 

si reducas 7 ad radicem cubam, quae est axis sperae, fient 343. 
Et scis quod cubus, id est quadratura eius, est tamquam 21 ad 
11 ad sperae quadraturam. Ideo multiplices 343 per 11, fient 
3773: partire per 21, evenient 179 2/3. Itaque radix cuba 179 
2/3 est latus cubi quod quaerebamus. 

The Trattato d'abaco, (fol. 117v. [1] ), on the other hand, 
presents the following methods of solution. 

Egl'è una spera che il suo diametro è 7 bracci; voglo de la 
sua quadratura fare uno cubo. Domando de' suoi lati. 
Quadra la spera che il suo diametro è 7, reca a radici cuba fa 
343; et, perché la spera è 11/21 del suo cubo et il suo cubo è 
343, perno montiplica 11 via 343 fa 3773, il quale parti per 
21, ne vene 179 2/3; però sirà il cubo radici cuba de 179 
2/3. Possiamo fare per altra via, dire così: sì conmo ài di sopra 
che il diametro de la spera ch'è radici cuba de 122 2/11 dà il 
lato del cubo radici cuba de 64, dunqua che te darà il diametro 
ch'è 7? Reca 7 a radici cuba fa 343, montiplica 64 via 343 fa 
21952, il quale parti per 122 2/11 ne vene 179 2/3, e la ra- 
dici cuba de 179 2/3 sirà il cubo per lato; sì conmo di sopra. 



In addition to ali this, further references to Euclid are 
found in the Libellus, Part III (nos. 23, 25, 27, 28), and 
there others to Archimedes (nos. 15, 16, 20, 23). 

The above citations from the two treatises show that Pie- 
ro does not reproduce problems already worked out in the 
Abaco in the Libellus without redefining their methods of so- 
lution and improving their accuracy. One cannot fail to notice 
that the Libellus is frequently more concise than the Trat- 
tato d'abaco and that it frequently adds passages to clarify 
mathematical procedures. At times, apparently dissatisfied with 
the form of solution utilized in the Abaco, Piero changes it 
in the Libellus. Frequently the resulting numerical answer 
differs. Sometimes, though the method of solution changes, 
Piero arrives at the same answer {Libellus, II, 30 and Tratta- 



54 Chapter Three 

to d'abaco, fol. 11 Ir [1] ). In other cases, where two methods 
of solution were given in the Abaco (for example, fol. 90r 
[ 1 ] ), Piero chooses to reproduce only the first in the Libellus 
(I, 28). But more often than not the problems in the Trattato 
d'abaco are executed with relatively little change in the Libel- 
lus. 

The consistent use of shorter sentences, more concise 
phrases, and more exact punctuation makes the whole of the 
Libellus, in contrast to the Abaco, appear clearer, more crys- 
tallized, and more ably constructed. Much of this impression 
must, however, be attributed to the translator of Piero's man- 
uscript, Matteo da Borgo, evidently a capable Latinist, for 
most likely the Italian manuscript he worked from read much 
like the Trattato d'abaco 8 . 

It would be a mistake, however, to conclude that Piero 
assembled the Libellus mainly from the earlier Trattato d'aba- 
co. Part III, entitled « Incipit tertia pars quinque corporum. 
De dimensione laterum ipsorum, divisionibus axis, superficie- 
bus, transmutationibusque unius corporis in aliud, etc, » 
represents entirely new work in exercises 1 through 13, and 
it reveals a new and penetrating study of Book XV of Euclid's 
Elements. This section, Piero 's « tertia pars », is introduced 
with a statement, which announces Piero 's intention to ex- 



8 Luca Pacioli, in his Summa de arithmetica, geometria, proportioni 
et proportionalità (Venice 1494), fol. 68v, reports that « maestro Matteo » 
translated the Prospectiva pingendi, and it is very likely that he also trans- 
lated the Libellus: « El sublime pie t or e ... maestro Piero de li Franceschi ... 
hane in questi di composto un degno libro de ditta prospectiva ... El qual 
lui feci vulgare: e poi el famoso oratore: poeta: e rhetorico: greco e latino 
(suo assiduo consotio: e similmente conterraneo) maestro Matteo lo recco a 
lengua latina ornatissimamente de verbo ad verbum: con exquisiti voca- 
buli. » For Matteo Cioni da Borgo, see L. Coleschi, Storia della città di 
Sansepolcro (Città di Castello 1886), p. 227; P. Farulli, Annali e memorie 
dell'antica e nobile città di S. Sepolcro (Foligno 1713), p. 64. See below, 
Appendix II, for an analysis of the respective Latin and Italian texts of 
the Libellus and Abaco. 



The « Libellus de quinque corporibus regularibus » 55 

plain how the sides of various bodies are related to one anoth- 
er and how the one may be contained within the other, and 
to treat the area of the sphere and its square, and stili other 
questions. 

Nunc in hac tertia parte volumus, sicut in principio dixi, ex- 
ponere quanta sint latera corporum unius ab altero contento- 
rum, et quot contineantur in uno, et quot in altero. Deinde 
dicam de superficie sperae atque quadratura, et de quibusdam 
axis divisionibus et superficiebus, et quadraturis a planis lineis 
factis, et transmutationibus sperae in cubos, et cuborum in spe- 
ram, et pariter sperarum in conos et conorum in speras 9 . 



The latter promise refers to exercises 14 through 29 of 
Tract III, on the sphere. The first part, describing the mea- 
surement of one regular body located in another, is found in 
the first thirteen exercises. In the Trattato d'abaco there are 
no exercises that measure one of the solids placed within anoth- 
er, nor are there references to Book XV of the Elements. 
Thus the first thirteen exercises of Part III of the Libellus 
represent a further stage in Piero's mathematical development. 

This work results from a consideration and elaboration of 
Book XV of the Elements, and this may be seen by examining 
two of Piero's exercises. No. 2 reads, « If a tetrahedron is pla- 
ced in a cube whose side equals 12 braccia, what is the side 
of the tetrahedron? » Piero solves the problem by first drawing 
diagonals through one face of the cube. « We know from Eu- 
clid, I, 47, [i.e., the Pythagorean theorem] », he writes, « that 
if the sides of a square equal 12, its diagonal will equal the 
square root of the squares of two sides added together », that 
is, V288. « And V288 will, furthermore, be the side of an 
inscribed tetrahedron according to Euclid, XV, 1 ». The first 



9 De corporibus regularibus, p. 540. 



56 Chapter Three 

proposition of Euclid, XV, 1, shows the relationship of a te- 
trahedron inscribed in a cube: « Intra propositum cubum cor- 
pus habens quatuor bases triangulas equalium laterum desi- 
gnare ». In this proposition a tetrahedron is inscribed in a 
cube so that each side of the tetrahedron equals the diagonal 
of each face of the cube. Hence knowing from Euclid, XV, 1, 
the relationship of the side of a cube to the side of a tetrahe- 
dron inscribed in it, Piero applied an arithmetical procedure. 
A second problem from Piero's Libellus, no. 3, reads, «If 
an octahedron is contained by a cube whose side is 12 braccia, 
what is the side of the octahedron? » To find its solution Pie- 
ro first places a tetrahedron, as in problem 2, in a cube, and 
its side is shown (as above) to be ^288. « Divide ^288 », 
Tie writes, « into four equal parts and you have ^72 ». And, 
« V 72 is the side of an octahedron contained in a cube whose 
side is 12 braccia ». This proposition, too, is demonstrated 
in Euclid, XV, 3: « Intra cubum assignatum figuravi odo ba- 
sium triangularium equalium laterum constituere cubo tnten- 
dimus inscribere octocedron » 10 . In proposition 3 the relation- 
ship between the side of a tetrahedron and that of an octahe- 
dron, i.e., the side of an octahedron is one fourth that of a te- 
trahedron, is explained. We read in Euclid. 

Constat enim ex ratiocinatione prima latera cuncta ipsius in- 
scripte pyramidis esse diagonos basium cubi. Et ex ratiocina- 
tione premisse liquet cunctos angulos octocedri in hac piramide 
distincti esse in lateribus ipsius piramidis. Quare manifestum est 
omnia angularia puncta huius octocedri esse in basibus assignati 
cubi. Igitur ex diffinitione habemus propositum. 

What Piero has, in effect, done in the first part of Part III 
is to apply arithmetical problems to the fixed geometrical pro- 



Opera, ed. Pacioli, fols. 141r-142r. 



The « Libellus de quinque corporibus regularibus » 57 

portions that exist among the regular bodies inscribed in one 
another as defined in Book XV of the Elements. In fact, ali 
of Piero's problems, save one, relate directly to Euclid's text. 
The following table outlines these correspondences exhaus- 
tively ". 

Libellus, III, no.: Elements, XV, proposition: 

1 . Octahedron in a tetrahedron 2 

2. Tetrahedron in a cube 1 

3. Octahedron in a cube 3 

4. Icosahedron in a cube — 

5. Cube in an octahedron 4 

6. Tetrahedron in an octahedron 5 

7. Cube in a dodecahedron 8 

8. Tetrahedron in a dodecahedron 10 

9. Octahedron in a dodecahedron 9 

10. Icosahedron in a dodecahedron 7 

11. Cube in an icosahedron 11 

12. Tetrahedron in an icosahedron 12 

13. Dodecahedron in an icosahedron 6 



2. Irregular bodies: The « Libellus de quinque corporibus », 
Part IV 

The exercises in Part IV of the Libellus, on irregular bod- 
ies, are mostly abstract and mathematical in nature, but a 
few can be related to the study of architecture. And, in their 
entirety, the problems relate to the study of perspective. The 



11 In the Divina proportione, Part I, Chapters 35-45, Pacioli demon- 
strates the proportional relationship of one regular body inscribed in another. 
With the exception of Book XV, prop. 2, an octahedron in a tetrahedron, 
his eleven examples follow Euclid's, even in their ordering. 



58 Chapter Three 

essential components of problems 2-9, 12-15, and 18, are found 
in the Trattato d'abaco. These exercises mainly measure irreg- 
ular geometrical solids formed from triangles, squares, hexa- 
gons, pentagons, and octagons, ali placed in a sphere. For 
example, if in a sphere whose axis is 12 braccia, Piero writes 
in exercise 6, an irregular body composed of four triangles and 
four hexagons is placed, what are the sides, surfaces and square 
of this solid? The Trattato d'abaco, fol. 107r-v, presents a 
similar problem. If, in a sphere whose axis is 6, an irregular 
body of four triangles and four hexagons be placed, what are 
its sides? And then, asks Piero, what is its square? 

The first problem in Part IV of Piero 's Libellus is, how- 
ever, one with which architects often dealt — the measurement 
of a 72-faced solid, a solid made up of twenty-four triangular 
and forty-eight four-sided polygonal faces, having unequal sides 
and angles. Piero writes that Johannes Campanus, the me- 
dieval translator and commentator of Euclid, taught how to 
construct this body in Book XII, proposition 14, of the Ele- 
ments. Though Piero does not mention its usefulness in ar- 
chitectural constructions, Luca Pacioli testifies to its impor- 
tance in this context in the Divina proportione, Part I, Chap- 
ter LIV, entitled « Del corpo de 72 basi piano, solido, e va- 
cuo » 12 . Half of this body, Pacioli writes, the half that resem- 
bles a hemisphere, is found in the « antico tempio pantheon ». 
A quarter of the solid is used in the architecture of a tribune, 
or apse, as in Milan, in the chapel of S. Satiro and in S. Maria 
delle Grazie: « E questo 72 basi molto doli architetti sia fre- 
quentato in loro dispositioni de hedifitii per essere forma asai 
acomodata maxime dove occurrese fare tribune o altre volte 
o vogliamo dire cieli » 13 . Pacioli's remarks on this topic re- 
late to the discussion of architectural representations in Pie- 



12 Divina proportione (Venice 1509), fol. 16r. 

13 Ibid. 



The « Libellus de quinque corporibus regularibus » 59 

ro's writings, for in the Prospectiva pingendi Piero furnishes 
directions for rendering, in perspective, « una cupola per ra- 
gione, la quale fusse commo uno quarto de una palla dal canto 
poncavo e fusse devisa in quadrati nelli quali fussero rosso- 

14 

ni » . 

Among the architectural elements which Piero showed 
how to foreshorten in the Prospectiva pingendi, there were 
also an eight-sided and a sixteen-sided column and a cross vault 
— « una volta in crociera » 15 (fig. 4). Similarly, Piero dem- 
onstrates in Part IV of the Libellus the correct mathematical 
measurement of a round column and a vault « per modum 
crucis » 16 (fig. 5). These three exercises relate to the architect's 
concern with architectural description, as well as to the paint- 
er's concern with pictorial perspective. The ability to measure 
vaults, columns, and other architectural parts entails knowing 
« la forza de le linee et degli angoli », necessary for perspective 
studies 17 . 

Concerning perspective, the loquacious Luca Pacioli is more 
informative about the connection between regular and irreg- 
ular bodies and perspective science than Piero is. When he 
translated and appended Piero's Libellus de quinque corpori- 
bus regularibus to his Divina proporzione of 1509, Pacioli in- 
serted the following lines between exercises 4 and 5 of Part 
IV of the Libellus, apologizing to the reader that the solids 
do not always appear in the margin of the treatise because 
they need to be drawn by one well practiced in the science of 



14 De prospectiva pingendi, ed. G. Nicco Fasola (Florence 1942), I, 202. 

15 Ibid., pp. 104, 112, 122. There are further architectural demonstra- 
tions in the Prospectiva pingendi regarding the base of a column (p. 148); 
the capital of a column (p. 157); a six-sided well (p. 106), and « uno tempio 
de odo facce » (p. 118). 

16 De corporibus regularibus, p. 571: « Est quaedam testudo, seu 
volta, per modum crucis, quae est prò qualibet facie 8 brachia, et in alti- 
tudine 4 brachia, tam in summitate arcuum quam in medio voltae: quae- 
ritur de superficie concava? » 

17 Prospectiva pingendi, p. 128. 



60 Chapter Three 

perspective. « Lee t or e non te maraviliare se de simili corpi 
composti de diverse e varie base non te se mette sempre in 
margine loro figure conciosia eh' le sieno dificilime farle in 
desegno pero che bisogna che sieno facte per mano de bono 
prospectivo quali non si posano sempre bavere... » 18 . 

For the most part the exercises presented in Part IV of 
the Libellus grew out of the subjects already treated in the Trat- 
tato d'abaco and the Prospectiva pingendi, and the relationship 
of the three treatises to one another is again closer than a super- 
ficial reading suggests. The same themes, such as the golden sec- 
tion and the regular bodies, are treated both in the Abaco 
and the Libellus, and the application of the measurement of 
regular polygons and polyhedra to painting is demonstrated 
in the Prospectiva pingendi. The exercises in ali three treati- 
ses derive from Piero's unifying interest in correctly measu- 
ring and correctly representing objects found in nature. Pie- 
ro's source, Euclid's Elements, remains moreover his life-long 
stimulus. 

The rules and methods for measuring two-dimensional poly- 
gons in Part I of the Libellus and three-dimensional polyhedra 
in Part II, as well as the definition of the relationship of one 
regular body to another and the exercises on the sphere in 
Part III, lay the groundwork for the more complex irregu- 
lar bodies discussed in Part IV. Many of these irregular sol- 
ids relate to the familiar objects in nature that fall to the 
painter to represent. Piero apparently considered the study 
of the progressively complex solids, such as those in the Abaco 
and Libellus, a prerequisite to the accurate representation of 
the more complex forms of nature. In this context, it is worth 
recalling the example of Albertus Elementa picturae, a mathe- 
matical treatise reworked for artists, which contains instruc- 
tions in drawing regular and irregular polygons, which in 



18 Divina proportione, fol. 22r. 



The « Libellus de quinque corporibus regularibus » 61 

turn constitute the rudiments of perspective. Thus it seems 
reasonable to see the same connection that Piero saw between 
his Libellus de quinque corporibus regularibus and his De pro- 
spettiva pingendi when he wrote to Guidobaldo hoping that 
his two works would be preserved side by side in some corner 
of the copious Montefeltro library, there to bask in the reflec- 
ted splendor of the young duke. 



Many of the figures presented in Part IV of the Libellus 
concern the measurement of building parts — columns, vaults, 
apses, domes, and the like. The correct representation of these 
was required of the architect in presenting his plans, of the 
painter in defining the architectural ambients for his histo- 
ries, and of the scenographer in designing and erecting his 
stage settings. Furthermore, it was of considerable consequen- 
ce for the punctilious designs of intarsia makers. 

Luca Pacioli's Summa arithmetica (which contained Pie- 
ro's geometrical exercises from the Abaco) was written for 
practitioners of the mechanical arts, among others, intarsia 
craftsmen, who needed to be well-versed in the rules of pro- 
portion that governed their compass and rule 19 . Later, in his 
Divina proportione, Pacioli discloses Piero's connections with 
intarsia workers, writing that the famous Lorenzo da Lendi- 
nara was « caro quanto fratello » to Piero . 



19 Summa arithmetica, Dedication to Guidobaldo: « De latti tutte 
Mecaniche. discorrendo in tutti exerticii. e mistieri. non si vede oculata 
fide. Che toltoli de mano la squadra. El sexto. con la lor proportione non, 
tanno che si peschino. Chi de tarsia, si nobilmente con tanta diversità de 
legnami per tutto apieno lunico. U.D. palaqqo ha disposto, se non le doi 
linee curva e retta: con suoi pontuali termini proportionata. » 

20 Divina proportione, fol. 23r: « e anco con quella prometto darve 
piena notitia de prospectiua mediami li documenti del nostro conterranea, 
e contemporale di tal faculta ali tempi nostri monarcha maestro Vetro de 
franceschi dela qual già feci dignissimo compendio e per noi ben apreso. 
E del suo caro quanto fratello Maestro Lorenqo canoqo da Lendenara: qual 



62 Chapter Three 

The principal motifs of early intarsia panels were usually 
simple objects of a geometrical character, or views of piazzas 
and buildings, rendered in perspective. In fact, the virtuoso per- 
spective techniques of the woodworkers of the Quattrocento 
resulted in the appellation « maestri di prospettiva », which 
was by and large the term applied to intarsia workers 21 . The 
substantial market they had to satisfy is reflected in the fact 
that there were in Florence alone eighty-four botteghe for in- 
tarsia and woodcarving in 1470 °. Moreover, fifteenth-century 
paintings, as well as household inventories, reveal that intarsia 
was of first importance for domestic furnishings. And in the 
ecclesiastical sector the situation was the same. The significant 
number of intarsia decorated choir stalls that have survived 
the ravages of time testify to their popularity. The simple 
designs, or patterns, for these works, handed from the mas- 
ter (often an architect or painter) to the crafsman, have not 
survived, but we can reconstruct some aspects of the appea- 
rance of such cartoons from the illustrations to the Prospectiva 
pingendi and the Libellus de quinque corporibus regularibus. 

For example, the mazzocchio in the Prospectiva pingendi 
(fig. 6) compares well with those found in the Lendinara choir 
stali in Modena (fig. 7); similarly, the hexagonal well on a base 



medesimamente in dieta faculta fo ali tempi suoi supremo chel dimostrano 
per tutto le sue famose opere di intarsia nel degno coro del Sancto a Pà- 
dua ....» 

21 G. C. Romby, Descrizioni e rappresentazioni della città di Firenze 
nel XV secolo, con la trascrizione inedita dei manoscritti di Benedetto Dei 
(Florence 1976), p. 73. In the «Memorie» of Benedetto Dei (Cod. Ric- 
cardiano 1853), fol. 90r, the intarsiatori of Florence are listed « Maestri 
di Prospettiva in Firenze tutti Fiorentini l'anno 1470. » He names fourteen 
adding, « et altri maestri che sono n° 19.» 

22 Ibid., p. 49, from Benedetto Dei's « Cronaca fiorentina, dal 9 di- 
cembre 1430 al 1480,» (Florence, Archivio di Stato, Manoscritti, no. 119), 
fol. 32v: « Florentia bella à 66 botteghe di speziali e à 84 botteghe di 
legnaiuoli di tarssie e 'ntagliatori. » 



The « Libellus de quinque corporibus regularibus » 63 

illustrateci in the Prospectiva pingendi (fig. 8) seems almost 
to be the model for the six-sided fonts on the seats of the Mo- 
dena stalls (fig. 9). The octagonal baptistery in Modena (fig. 
11), too, grows out of the «tempio de odo facce» shown fore- 
shortened in the Prospectiva pingendi (fig. 10). As a last ex- 
ampie, one may compare the illustration o£ the 72-faced solid 
in the Libellus (fig. 12) and the cupola in the Prospectiva 
pingendi (fig. 13), with the intarsia view of a choir in Monte 
Oliveto by Giovanni da Verona (fig. 14). The same master 
translated the regular and irregular solids invented by Piero 
(and published by Pacioli, figs. 20, 21), into skillful wood 
intarsia panels in S. Maria dell'Organo in Verona (fig. 22). 

This illusionary world of scientific tools, musical instru- 
ments, sacred and secular still-life objects, set beside Street 
perspectives and in conjunction with purely stereometrical bod- 
ies, represents the triumph of a conception of space based on 
numerical and geometrical rules. These fifteenth-century in- 
tarsia panels are, in a way, visual realizations of the mathe- 
matica! exercises in Piero's treatises. Both in what they repre- 
sent and in their methods of execution they are an emblem of 
the union of art and science in the Renaissance. 



IV. 

Applications of the Art of Measurement 



Piero's stereometrie conceptions and inventions were first 
made widely available to artists and treatise writers through 
Luca Pacioli's publication of them. In 1494 Pacioli appended 
the stereometrie exercises in the Trattato d'abaco to his Stim- 
ma de arithmetica, geometria, proportioni et proportionalità, 
and in 1509 he published the Libellus de quinque corporibus 
regularibus in Italian at the end of his Divina proportione. In 
neither case did he acknowledge Piero's authorship ! . Since 
most of Piero's manuscript writings eventually disappeared 
from view, Pacioli's « plagiarism » was essential to the wide- 
spread dissemination of Piero's ideas, and it sowed the know- 
ledge of geometrie solids that became instrumentai in the de- 
velopment of perspectival and proportional exercises in Italy 
and in the North. Indeed the theme of the regular bodies be- 
came a standard feature of later Renaissance treatises. 

Although fundamental, the relationship of the abstract 
planar figures and the three-dimensional solids of Piero's mathe- 
matical treatises to his more famous perspective book is not 
as immediately apparent today as it was in the fifteenth and 
sixteenth centuries. Yet, as we saw, the taciturn Piero himself 



1 See below, Appendix I. 



Applications of the Art of Measurement 65 

implicitly Telateci the study of regular bodies to the study of 
perspective when he requested that the Prospectiva pingendi 
and the Libellus de quinque corporibus regularibus be shelved 
together in the Urbino Library. 

It was Luca Pacioli who made explicit the intrinsic con- 
nection that exists between the regular and irregular solids, 
on the one hand, and perspective, on the other. In his Stim- 
ma arithmetica Pacioli lists artists with whom he had discussed 
the science of perspective — « Quali sempre con libella e 
circino lor opere proportionando a perfection mirabile condu- 
cano » 2 And he recommends Piero's Prospectiva pingendi, 
a « dignissimo compendio, e per noi ben apreso », to readers 
of both his own printed works 3 . 

Then in his Divina proportione Pacioli included fifty-nine 
full-page woodcuts of regular and irregular bodies drawn in 
perspective and based on models prepared by Leonardo da 
Vinci 4 . These geometrical polyhedra were rendered in solid, 
hollow, and steUated variations and accompanied by their prop- 
er Greek and Latin names. Immediately following these 
illustrations, Pacioli translated and appended Piero's Libellus 
de quinque corporibus regularibus, without identifying its 
author. To the translation, however, Pacioli added a paragraph 
of his own apologizing for not including in the margins illus- 
trations of the geometrical solids he discussed, but these, he 
wrote, must be drawn « per mano de bono prospettivo >> , 
and therefore he referred his readers back to the preceding 
designs of Leonardo. In so doing, Pacioli not only joined the 
study of perspective to his own work on regular bodies, but 



2 Summa de arithmetica, geometria, proportioni et proportionalità (Ven- 
ice 1494), Dedication. . 

3 Summa arithmetica, fol. 68v; Divina proportione (Venice 1509), tol. 

23r. 

4 See above, Chapter I, n. 14. 

5 Divina proportione, fol. 22r. 



Chapter Four 

he also connected the geometrical exercises contained in Pie- 
ro's Libellus to lessons in perspective. 

Vasari, of course, had perceived an immediate connection 
between the mathematics of Piero and his painting, joined as 
they were through the science of perspective. For Vasari, Pie- 
ro was a past master of regular bodies as well as of arithmetic 
and geometry, and his accomplishments in these sciences are 
illustrated by ali his works, which are full of perspectives. But 
one work in particular shows this better than any other, a lost 
painting, or perhaps better, a drawing of a « vaso » shown in 
perspective. From Vasari's account it is clear that Piero 's draw- 
ing utilized methods of constructing and foreshortening per- 
fect curves on the basis of regular bodies, that is, methods 
founded in his mathematical studies. This is what Vasari says 
about the work: « ...un vaso in modo tirato a quadri e facce, 
che si vede dinanzi, di dietro e dagli lati, il fondo e la bocca: 
U che è certo cosa stupenda, avendo in quello sottilmente ti- 
rato ogni minuzia, e fatto scortare il girare di tutti quei circoli 
con molta grazia » 6 . 

Understandably, this striking description has been asso- 
ciated with a drawing in the Uffizi of a faceted chalice com- 
posed of circles, polygons, and mazzocchi set in perspective, a 
drawing usually attributed to Paolo Uccello 7 . It recalls, too, 
Vasari's description of some of Uccello's favorite perspective 
exercises — « mazzocchi a punte a quadri tirati in prospettiva 
per diverse vedute, e palle a settantadue facce a punte di dia- 
mante, e in ogni faccia brucioli avvolti su per e bastoni » 8 — 
nothing less than solid, hollow, and stellated polyhedra drawn 
in perspective, exercises that entailed the same kinds of proce- 
dures as Piero 's vase. 



6 Vasari-Milanesi, II, 491. 

7 J. Pope-Hennessy, Paolo Uccello, Complete Edition, 2nd ed. (Lon- 
don 1969), p. 27, piate 86. 

8 Vasari-Milanesi, II, 205. 



Applications of the Art of Measurement 67 

The rapici dissemination of Piero's ideas and inventions 
was, to an extent, ensured by the public for whom Pacioli wrote 
his books. In the Summa arithmetica Pacioli addressed stu- 
dents of architecture and perspective, as well as designer s of 
tarsie, and in his Divina proportione, « ciascun studioso di 
Philosophia, Prospectiva, Pictura, Sculptura, Architectura, Mu- 
sica e altre Mathematice » 9 . Moreover, Piero's inadvertent pub- 
licist took pride in his friendships with artists. Among the 
many artists he engaged in long discussions over the science of 
perspective and proportion, Pacioli counted, in Venice, Genti- 
le and Giovanni Bellini « carnai fratelli », and, in addition, a 
certain Hieronimo Malatini prospectivo, and elsewhere, Andrea 
Mantegna, Melozzo da Forlì, and Marco Palmezzano, and of 
course Piero, ali names synonymous with perspective 10 . No 
doubt these artists and others sought out Pacioli for his advice 
on the scientific questions of measurement and proportion 
taught in his treatises. And, Pacioli had earlier taught arith- 
metic and geometry to the scalpellini and architects of Sansepol- 
ero, for whom he composed a Vitruvian architectural text n . 

•k * * 

Certainly one further artist whom Pacioli knew well was 
the painter of the impressive portrait of Pacioli and of the 
world that occupied his mind, a painting long in Urbino and 
now in Naples, which bears the inscription, « IACO. BAR. 
VIGENNIS P. 1495 », identifying its artist and its date (fig. 
15). At a glance, the work seems almost a manifesto promoting 
an ongoing dialogue between art and mathematics. And, exa- 
mined closely in its details, it reveals itself to be an accurate 



9 Divina proportione, title page. 

10 Summa arithmetica, Dedication. 

11 See above Chapter I, n. 12. For Pacioli, see H. Staigmuller, 
« Lucas Paciuolo, eine biographische Skizze, » Zeitschrift fiir Mathematik 
und Physik, XXXIV (1889), 81-102, 121-28. 



68 Cbapter Tour 

illustration of Pacioli's artistic-mathematic colloquies, an illus- 
tration that indeed preserves, if not the substance of past 
conversations, at least the echoes of their essential themes. 

In his portrait, now in Naples, Pacioli stands, as if in an 
aula before his artistic acquaintances and ali the rest who 
would assemble to learn. Though dressed in his Franciscan ha- 
bit, it is Pacioli the learned teacher of mathematics who con- 
fronts us. His table is laid out with ali the trappings of his pro- 
fession, the instruments and accessories of his various róles — 
teacher, mathematician, translator, and scientific publicist. 
With one hand Pacioli indicates a passage in the book opened 
before him, and with the other, he illustrates its words with 
a geometrie diagram drawn on a slate, inscribed « EVCLI- 
DES » along the side of its frame. Before him, a heavy, mag- 
nificently bound tome bears the letters, « LI. R. LVC. BVR. », 
which identify it as work by Pacioli — « Liber Reverendi 
Lucae Burgensis ». On top of this book stands a dodecahedron 
made of wood, and to Pacioli's side a transparent crystal icosa- 
hexahedron is suspended. Across the table are scattered his 
chalk and sponge, his compass and protractor, and his pen, 
ink, and case, as it were, the tools of his trade. 

Old inventories, the first in 1582, identify the elegant 
young man behind Pacioli as the Montefeltro prince, Guido- 
baldo 12 . This identification certainly results from an inscrip- 
tion that once ran along the bottom of the panel — « Divo 
principi Guido » — which a busy inventory compiler appar- 
ently took to mean that Guidobaldo was represented in the 
painting 13 . « Divo principi Guido » is, of course, a dative, and 



12 See F. Sangiorgi, ed., Documenti urbinati: inventari del Palazzo 
Ducale, 1582-1631 (Urbino 1976), pp. 40, 57, inventories of 1582 and 1599. 

13 Ibid., p. 205, inventory of 1631: «Un ritratto di un frate di San 
Bernardino con un giovane apresso vestito di peliccia all'antica: segnato 
al basso Divo principi Guido, in tavola. » 



Applications of the Art of Measurement 69 

it records the dedication of the painting to the young prince. 
In much the same manner Pacioli had the year previous dedi- 
cated the Stimma to him. Indeed the portrait bears no resem- 
blance to authentic likenesses of the prince. Given the turn of 
the head, the side-long glance, and the age of the painter of 
the work (« vigennis »), it is tempting to believe that we see 
the artist's own self -portrait 14 . 

If we examine his picture more closely, we will see that 
the artist's mirror reflects not just the stage properties of Pa- 
cioli's classroom, but his lesson as well. On the opened pages 
in the bound manuscript that Pacioli takes as his text, only 
the words « LIBER XIII » can be made out. « LIBER XIII » 
can only refer to the last book of Euclid's Elements, the locus 
classicus of the regular bodies. The diagrams in Pacioli's book 
are legible, and, by comparing them with Pacioli's later edition 
of Euclid, we see that Pacioli is consulting his manuscript of 
Euclid's Elements. Indeed, he points to Proposition 8 of Book 
XIII, « Omnis trianguli equilateri quod a later e suo quadra- 



14 Althought not visible in the Alinari photograph (34011), the young 
man in the Naples painting wears a rounded cap (see B. Molajoli, Il Mu- 
seo di Capodimonte [Naples 1961], pi. XVII) of a common type, which is 
also encountered among Vasari's portraits of artists. For instances of similar 
headgear, see W. Prinz, Vasaris Sammlung von Kùnstlerbildnissen (Flor- 
ence 1966), p. 107 (Carpaccio), and Nos. 23 (Lippo), 40 (Giuliano da 
Majano), 41 (Piero), 44 (Lazzaro Vasari), 59 (Ercole de' Roberti), 60 
(Giovanni Bellini), 69 (Verrocchio), and 71 (Filippino Lippi). The 
Naples youth's fine costume is reproduced in nearly every detail in an appa- 
rently contemporaneous Venetian portrait of a Youth with Fur in Venice 
(Museo Correr), variously attributed to Gentile Bellini, Giovanni Bel- 
lini, Carpaccio, and Mocetto (G. Perocco, Tutta la pittura del Car- 
paccio [Milan 1963], n. 84 and pi. 192), indicating possibly a Venetian 
localization of the costume. The extravagant hairstyle of the Naples por- 
trait may be again compared with Vasari's artist portraits; note particularly 
a self -portrait of Ercole de' Roberti reproduced in Prinz, p. 51. Three 
authentic portraits of Guidobaldo are in Vienna, Kunsthistorisches Mu- 
seum, Ambraser Coli.; Florence, Uffizi; and in B. Castiglione, Il Corteg- 
giano, Rome, Vatican, MS. 



70 Chapter Four 

turn describitur triplum est quadrato dimidii diametri circuii 
a quo triangulus ipse circumscribit » 15 , which states, « If an 
equilateral triangle be inscribed in a circle, the square on the 
side of the triangle is triple the square on the radius of the 
circle » 16 . This theorem is fundamental to the inscription of 
regular bodies in a sphere, for it establishes the proportional 
relationship of the equilateral triangle to a circle, a correspond- 
ence required for the eventual construction and inscription 
of the tetrahedron, octahedron, and icosahedron. In modem edi- 
tions of the Elements this proposition is numbered 12 and im- 
mediately precedes the proposition concerning the construction 
of the simplest regular stereometrical figure, the tetrahedron, 
and its proportional relationship to the sphere. Looking then 
to Pacioli's slate, shown in perspective, we see that he has 
copied from the margin of his Euclid the diagram of this theo- 
rem (Proposition 8), a triangle inscribed in a circle, and he 
demonstrates the proof of the theorem to his listeners, who, we 
understand, are learning the inscription of regular bodies in a 
sphere " — a centrai theme from Piero 's Abaco that was re- 



15 Euclidis Megarensis philosophi acutissimi ... Opera, ed. L. Pacioli 
(Venice 1509), fol. 124r-v. 

16 The Thirteen Books of Euclid's Elements, ed. T. L. Heath, 2nd 
ed. rev. (New York 1956), III, 466-67. 

17 The line that Pacioli has just constructed and now indicates with 
his pointer, when continued to the circumference, will be equivalent to the 
radius of the circle into which the equilateral triangle is inscribed, and it 
is upon this element that the proof of the theorem depends. To demonstrate 
the proof, as may be seen on Pacioli's slate, the inscribed equilateral triangle 
needs be bisected, and the bisecting line continued to the far side of the 
circumference of the circle. If then its point of intersection with the cir- 
cumference be joined by a straight line to one of the base vertices of the 
triangle, it will be seen that the resulting line is one sixth of the entire 
circumference, or one side of a hexagon, and hence equal to the radius 
of the circle. From this recognition, through a series of simple logicai steps, 
one rapidly arrives at the conclusion, i.e., the square on the side of the 
triangle is triple the square on the radius, and thus the proposition is 
demonstrated. Since Pacioli's slate is sharply foreshortened, the construc- 



Applications of the Art of Measurement 71 

produced in the regular bodies treatise in Pacioli's Summa de 
arithmetica, geometria, proportioni et proportionalità, pub- 
lished at Venice in 1494. This small treatise was, like the paint- 
ing itself, dedicated to Guidobaldo da Montefeltro. The Sum- 
ma, a compendium of nearly 300 folios, is the book we see at 
the right, inscribed with Pacioli's name, for in 1495 it was 
his only printed work. 

But Pacioli's publications did not come to an end with the 
Summa. His lessons, too, continue, and the right hand page 
of his copy of Euclid shows what lies ahead. Here the diagram 
indicates another theorem fundamental in the development of 
Pacioli's mathematics, Proposition 9, concerning a line divided 
according to the golden section, or « divina proportione »: 
« Si latus exagoni equilateri latusque decagoni equilateri quos 
ambos unus idemq circulus circumscribit sibi invicem in lon- 
gum directumq. Coniungantur totalinea ex eis composita s.m. 
proportionem habentem medium et duo extrema divisa erit 
maiorq. eius portio latus exagoni » 18 . Modem geometers ren- 
der Euclid thus: « If the side of the hexagon and that of the 
decagon inscribed in the same circle be added together, the 
whole straight line has been cut in extreme and mean ratio 
and its greater segment is the side of the hexagon » 19 . Illus- 
trated in the margin of Pacioli's manuscript is a circle with 



tion of this line has been begun and is prominently displayed in the left 
forvvard segment of the circle, rather than in the far segment, nearest Pa- 
cioli, where it must first be constructed in order to make the proof. It has 
been displaced for the sake of clarity, and it is easy to see why, if one 
notes that ali three segments defined by the inscribed triangle are, in 
objective terms, perfectly congruent, and hence of equal size. Subjectively, 
however, rendered perspectivaUy, the surface area of the far segment has 
become insufficient to permit the legible indication of the radius in its 
proper place. 

18 Euclid, Opera, ed. Pacioli, fols. 124v-125r. Pacioli teaches this 
again in the Divina proportione, Part I, Chapter 16 

19 The Thirteen Books, ed. Heath, III, 455-57. 



72 Chapter Tour 

the side of a decagon inscribed in it and, added to it, the side 
of a hexagon (equal to the radius of the circle). 

The last illustrations on the manuscript folio before Pa- 
cioli represent the di vision of lines, and these undoubtedly 
relate to the lines shown on Pacioli's slate. Knowledge of the 
proportional ratios of both rational lines and lines divided ac- 
cording to the golden section was necessary for proposition 9 in 
Euclid's book, for the side of a decagon is an irrational line 
and the side of a hexagon is rational. Pacioli discussed linear 
proportions extensively in his commentary to Book V of the 
Elements, and he prefaced this book with the text of a lecture 
he gave on proportion in the church of San Bartolomeo in Ven- 
ice in 1508 20 . The division of lines into their mean and ex- 
treme ratios is, furthermore, the subject of propositions 1 
through 6 of Book XIII. The lines illustrated on Pacioli's slate 
then play a distinct róle in the larger theme of the painting, 
the Euclidean-Platonic solids. 

In the lower left corner of Pacioli's slate, a column of 
square roots is summed, and square roots, or irrational numbers 
are required to solve exercises related to the golden section. 
Two of the numbers, 621 and 925, are remarkably dose to 
numbers in the Fibonacci series, which in turn, approximate 
most closely the golden section proportion 21 . 

The golden section is also a prerequisite for constructing 
the dodecahedron, represented in perspective to the right. In 
turn, the dodecahedron is the one regular body which subsumes 
in its construction the other four. Thus it stands as a sym- 
bol of them ali, and of Pacioli's lessons, for it unites the two 
principal motives we associate with his name, the regular bod- 



20 Euclid, Opera, ed. Pacioli, fol. 31r-v. 

21 The numbers in the Fibonacci series run: 1, 2, 3, 5, 8, 13, 21 ... 
610, 987, 1597 ... See E. Colerus, Piccola storia della matematica da 
Pitagora a Hilbert (Turin 1962), pp. 134-35. 



Applications of the Art of Measurement 73 

ies and the « divina proportione », and it visualizes the im- 
mediate derivation of these themes in Piero's mathematics °. 

It was nearly fifteen years later, 1509, when Pacioli's Di- 
vina proportione was finally published in Venice. The manu- 
script, however, was completed in 1497 ° and thus, at the 
time the Naples painting was executed, Pacioli was already 
composing it. The lessons of Pacioli's portrait, by linking the 
golden section and the regular bodies, anticipate the develop- 
ment from the Summa to the Divina proportione. 

The most impressive feature of Pacioli's portrait remains 
to be noted. It is the shimmering, celestial crystal icosahexa- 
hedron, suspended to Pacioli's right, where, though im- 
mersed in light, it remains isolated by its singular perfection. 
Through the painter's perspective vision, it is at once both 
solid and hollow, in itself an extraordinary display uniting 
perspectival skills to the measurement of stereometrical solids. 
AH its shining, transparent f aces — eighteen squares and eight 
equilateral triangles — are simultaneously visible and reflect 
buildings, apparently the Palazzo Ducale in Urbino, from an 
unseen open window. 

In his Divina proportione Pacioli describes the construc- 
tion of just such an icosahexahedron. The chapter, number 53, 
is entitled « Dela formatione e origine del corpo del 26 basi, 
piano, solido, over vacuo, e delo elevato solido, over va- 
cuo » 24 and illustrates a solid, hollow, and stellated example 
of this form (fig. 20). The text of the Divina proportione ex- 



22 Pacioli wrote in the Divina proportione, Chapter V, « El quale ... 
senza la nostra proportione non e possible poterse formare. » And in the 
same chapter he wrote concerning measurement of ali five of the regular 
bodies: « Li quali 5 regulari non è possible fra loro poterse proportionare. 
ne dola spera poterse intendere circonscriptibili senza la nostra detta propor- 
tione. » 

23 Pacioli, Divina proportione, fol. 23r. 

24 Ibid., fol. 15v., and table of contents. 



74 Chapter Four 

plains that the two most important solids for architects are the 
icosahexahedron, or 26-faced solid, and the 72-faced solid, la- 
belled by Pacioli « Hebdomecontadissaedron » (fig. 21). Of 
the 26-faced solid, Pacioli writes: « E sia la sua scientia in 
molte considerationi utilissima a chi ben la acomodare maxime 
in architectura » 25 . 

The detailed, closely focused mathematical programme of 
Pacioli's portrait was, no doubt, spelled out by Pacioli to his 
painter, who in turn must have known, or quickly learned, the 
mathematical concepts he was called upon to represent. Pacioli, 
an old hand with artists and an uninhibited pedagogue, clearly 
intervened in the painterly disposition of his ideas, for instance, 
in the studied positioning of the stereometrie bodies in or- 
der to show their faces to best advantage, to cast their sha- 
dows most effectively, and to diminish most tellingly in per- 
spective. And the realization in perspective of the rather in- 
tricate demonstration on Pacioli's slate evidences a direct col- 
laboration between the painter and his intellectual mentor. 

Pacioli apparently went far beyond even this in setting 
the confines of the visual programme. His Summa, published 
the previous year contains a small recurrent woodcut mono- 
gram (« LU ») incorporating the friar's portrait (fig. 16). In 
it, Pacioli, tonsured and wearing his Franciscan habit, stands 
before a table, pointing with a compass to the figures of a 
circle and a triangle in an open book before him, teaching as 
it were from the pages of the Summa, teaching, it almost seems, 
the same problem as in the Naples painting. This portrait 
simply expands the concept and format already established in 
the monogram. Pacioli's róle in the design of the painting does 



25 Ibid., fol. 15v. In the Summa arithmetica (part II, fol. 68v) Pacio- 
li speaks proudly of the geometrical bodies he constructed for Guidobaldo 
da Montefeltro, Duke of Urbino; Pietro Valletari of Genoa, Bishop of 
Carpentras; and Giuliano della Rovere, the Cardinal of S. Pietro in Vincoli. 



Applications of the Art of Measurement 15 

not, perhaps, end here. It is as if Piero had set his stamp on 
the design of Pacioli's stiff, heavy robe, which fills the center 
of the painting. Pulled in at the waist by a knotted cord, with 
large, weighty tubular folds moving from above and below 
toward the center (almost like lines drawn upon two intersec- 
ting cones), the robe repeats with clear understanding a com- 
mon formula of Piero, a stylization that we meet in infinite 
variations turning the pages of any book on the painter. This 
stylization is little dependent upon the forms of the religious 
habit Pacioli wears, and it does not seem particularly congru- 
ent with the Naples painter's style, as far as it can be observed 
in this picture. It does, however, echo the monogram portrait 
of Pacioli, and the mathematician, himself, may have had in 
his possession a drawing, or some other record of his own 
likeness from the hand of Piero, inasmuch as Piero had por- 
trayed him some twenty years earlier in the Brera altarpiece 26 . 
Certainly Pacioli required, or at least approved of this 
quasi-quotation of Piero's style, perhaps understanding it as 
a tribute to the recently deceased master whose ideas he 
propagated. 

In any event, we must recognize Pacioli's responsiblity 
in the design of his own portrait. In the broadest possible 



26 In his Vite de matematici (1589), B. Balbi describes a portrait of 
Pacioli in the Urbino Guardaroba by Piero della Francesca, which, how- 
ever, is very possibly to be identified with the Naples work (Battisti, 
Piero della Francesca, II, 95, 257). F. Heinemann {Giovanni Bellini e i 
belliniani [Venice 1963]), I, 274-75, wrote that the « ritratto di Fra' Luca 
fu ideato derivandolo dal prototipo di Piero della Francesca. » M. Meiss's 
contention, reiterated by K. Clark {Piero della Francesca, 2nd ed. [Lon- 
don 1969]), p. 68, that Pacioli, a Franciscan friar, could not conceivably 
have been represented as the Dominican Peter Martyr in the Brera Aitar 
is not correct. The Brera painting was executed by 1474, and the first notice 
we have for Pacioli as a friar is in 1477. See Staigmuller, « Lucas Paciuo- 
lo. » p. 87; cf. A. Fantozzi and B. Bughetti, « Il terz'ordine francescano 
in Perugia dal sec. XIII al sec. XIX, » Archivium Franciscanum historicum, 
XXXIII (1940), 357-58. 



76 Chapter Four 

sense the invention is his. And in his hands the youthful, 

twenty-year-old painter was no doubt easily molded, and 

perhaps Pacioli hoped that under his tutelage, he would be- 
come a new Piero. 

Who then is the painter of the work, or who is IACO. 
BAR. VIGENNIS, who painted it in 1495? When identified 
with a known artist, IACO. BAR. is always said to be Jacopo 
de'Barbari, although this identification has for many reasons, 
some of them rather futile, often been doubted by scholars. 
Serious chronological objections to the identification have, in 
recent years, been effectively dismantled by Creighton Gilbert, 
who also showed how intrinsically unlikely ali the possible 
alternatives to the identification are v . Nonetheless, it is stili 
sometimes felt that on stylistic grounds the painting is diffi- 
cult to reconcile with the rest of Barbari's oeuvre. 



27 C. Gilbert, « Alvise e compagni, » Scritti di storia dell'arte in onore 
di Lionello Venturi (Rome 1956), I, 290-92, n. 6; idem, « Ancora di Ja- 
copo de' Barbari,» Commentari, Vili (1957), 155-56; idem, «When Did 
a Man in the Renaissance Grow Old? » Studies in the Renaissance, XIV 
(1967), 7-32; idem, «Barbari, Iacopo de',» Dizionario biografico degli Ita- 
liani, VI, 44-46. 

Though the inscription shows signs of repainting, it is wrong to ques- 
tion its authenticity (Heinemann, Bellini, p. 274 and J. A. Levenson in 
Early Italian Engravings front the National Gallery of Art [Washington, 
D. C. 1973], p. 344, n. 15; cf. Battisti, Piero della Francesca, II, 95; 
A. De Rinaldis, Pinacoteca del Museo Nazionale di Napoli: catalogo 
[Naples 1911, pp. 111-12]). The painting, acquired for Capodimonte in 1903, 
apparently never passed through the art market. A late, « spurious » inscrip- 
tion would not pin down the date of the painting so correctly, nor would 
it seek to ascribe the work to the then not so well known Jacopo de' Bar- 
bari, who had no known connection with Pacioli and whose paintings are 
not very distinguished, nor would it identify him so cryptically and with 
the unusual qualification « vigennis ». The puzzles of the inscription are 
authentic; they were not contrived by a later tamperer. Furthermore the 
lettering is consistent with that on the book as well as with Pacioli's 
own alphabet in the Divina proportione. Compare especially the rather dis- 
tintive R with a tail. 



Applications of the Art of Measurement 77 

To answer this objection, we must take into account sev- 
eral characteristic aspects of Barbarie artistic personality and 
keep in mind a few peculiarities that we have noticed concern- 
ing Pacioli's portrait. Although Barbari has a number of 
distinctive traits, as an artist he did not always show a very 
strong or decided character. Moreover, he was an uneven, 
though sometimes quite distinguished performer; at times his 
works are eclectic and fragmentary from a stylistic point of 
view; finally he seems to have been extremely impressionable. 
Thus very many things may be possible in his paintings and 
indeed they are to be found in his autograph works alone. Yet, 
aside from the panoramic map of Venice, next to nothing he 
executed in Italy has been identified. And many of the works 
Barbari executed in the North are so filled with Germanisms 
that it was long believed he was not an Italian at ali 28 . We 
cannot expect to find these elements in his Italian works. 

On the other hand, we have seen Pacioli's large ròle in 
establishing the format of his portrait, and the prescribed, in- 
evitable shapes of the geometrical objects further limit the 
sphere in which the style of « Jaco. Bar. vigennis » can mani- 
fest itself . Practically, one may look best to the portraits, the 
young man's costume, and the still-life objects. 

In these circumstances, it is not really necessary to dem- 
onstrate a stylistic identity with Barbari's works executed 
after he left Italy. And rather than looking for a clear transi- 
tion across this cultural divide, it is probably sufflcient to find 
points of stylistic contact to add to the elements of the culture 
of Pacioli's portrait which recur in Barbari's later biography 
and art, and to add this to the common stylistic tradition from 
which both « Jaco. Bar. » and Jacopo de'Barbari are usually 
said to derive, a Bellinesque tradition — late Antonello, late 



28 Levenson, Italian Engravings, pp. 341-42. 



78 Chapter Four 

Alvise Vivarini, late Quattrocento Venice. In doing so it seems 
correct to work, in part, with externals, Nonetheless bridges can 
be built across the Alps, and they need only confimi what is 
a securely signed and dated work. 

Only a handful of signed, unquestionably authentic paint- 
ings belong to Jacopo — we can count only eight 29 . Nearly 
ali of them show some variation on the quasi-Flemish realism 
of the Naples painting. It is also noteworthy that of these 
eight, six nave the « sfondo nero » of the Naples portrait, a 
taste for dark backgrounds and silhouetting which appears in 
various guises in Jacopo's work. And, too, the bust or half- 
length format is that found in most of these works. But, there 
are more specific links in terms of details. Pacioli's pudgy, 
slightly clumsy hand with its poorly drawn nails reappears in 
Barbari's portraits of S. Osvaldo (1500; Private Coli., Hun- 
gary), that of Albrecht of Brandenburg ( 1508; Coli. H. Kisters, 
Kreuzlingen), and that of a Young Man in Vienna. The last 
of these portraits has apparently lost much of its surface, but 
the strong drawing of the eyes, the forceful modelling of the 
nose, and the rather full lips which seem lightly pressed upon 
the face are ali immediately analogous to Barbari's possible 
self -portrait in Naples. These features are repeated, along 
with the heavily drawn brows, in the portrait of the Margrave 
Albrecht, where the note of coarseness common to the Vienna 
and Naples portraits is understandably muted. The thin-lipped, 
solemn likeness of Pacioli recurs in the portrait of S. 
Osvaldo, and the flesh of his face similarly disintegrates 
under the weight of his years. The same phenomenon is 
represented in a wholly diflerent style in the signed Johnson 
panel (Philadelphia, 1503) and in the Louvre Sacra Con- 



29 Gilbert, Dizionario biografico, pp. 44-45. For the portrait of Al- 
brecht of Brandenburg, see the recent exhibition catalogue, D. Koep- 
plin and T. Falk, Lukas Cranach, I (Basel, 1974), no. 4. 



Applications of the Art of Measurement 79 

venazione. « Jaco. Bar. » describes the corkscrew curls of 
the Naples youth with sets of concentric, circling gold lines, 
and Jacopo de'Barbari employs the same device in his bust 
of Christ in Weimar and in the St. John Baptist in his painting 
in the Louvre. Moreover, Barbari employs several quite dis- 
tinctive configurations of draperies. It makes an amusing game 
to trace their reappearances in his paintings and engravings. 
The foldings of Pacioli's garment, as we have seen, need not 
be expected to appear among them, although the large, circling 
openings of his sleeves belong to a taste also present in the 
Louvre Sacra Conversazione. As an index of « Jaco. Bar. » in 
this respect, we must examine principally the sleeve of the 
standing youth. It happens that the close-fitted, crumpled 
sleeves of the Madonna in Jacopo's Sacra Conversazione now 
in the Berlin Museum (neither signed, nor dated, but always 
accepted) are formed in an identical pattern of alternating 
ridges and needle-eye depressions. This coincidence is not for- 
tuitous, since it repeats itself in the Virgin of the Louvre 
painting, in a drawing of a male allegorical figure in the Uffi- 
zi, and in a few of Barbari's prints. 

Jacopo de'Barbari's best known and most highly regarded 
painting, the Munich Stili-li fé (1504), resembles Pacioli's por- 
trait in Naples more closely that it does any other of Barbari's 
signed paintings. Its « sfondo bianco » is the aesthetic equi- 
valent to the dark backgrounds of the Naples painting and 
of other works by Jacopo de'Barbari. Both are signed on an 
unfolded slip of paper, set realistically, trompe-V oeil , in light 
and shadow. Both are concerned with textures and describing 
surfaces: the fur collar and feathers are treated with a similar 
touch and a similar sense for mottled effects, and the chain 
links of the mail match the metallic curls of the standing 
youth. Ali these realistic surfaces are infused with the same 
pristine light, and the still-life objects, whether a duck's foot 
or an inkpot, ali cast carefully calculated shadows. The hard, 
metallic edges that exactingly outline the openings of the hood 



80 Chapter Four 

and sleeves framing Pacioli's head and hands were created by 
a mind that would delight in the precise, steely circles of the 
metal armbands silhouetted against the shadows and the light 
ground of the Munich panel. The armor is, like the crystal 
icosahexahedron, a variation on a geometrically regular body, 
and the silvery facets of both are struck by sudden refracted 
light, often broken into reflected colors, which streak across 
the surfaces and which run along the edges. The same geome- 
trical ordering of light and shade that defines the surface of 
the wooden dodecahedron in Naples forms the hunting spear 
in Munich, and the depiction of the wood grain of the dodec- 
ahedron is paralleled in the background of the Munich paint- 
ing. Thus the luminous, optical, perspectival, geometrical 
culture that Pacioli's portrait proclaims, inconspicuously, but 
nonetheless surely, comes to fruition in the Munich still-life. 
This culture may also be traced in Barbari's graphic works, 
for instance, in his penchant for the bird's-eye view, a pano- 
ramic perspective; in the scorci of his Venice Neptune, his 
Three Captives, and his St. Sebastiana in the geometrical stag- 
ing of his Presentation, Guardian Angel, and Two Philoso- 
phers; or finally in the crystalline celestial sphere in his Apollo 
and Diana, itself a graphic experiment with radiance, optical 
transparency, and eccentric perspective. 

AH these bridges that we have seen between « Jaco. Bar. » 
and Jacopo de'Barbari, are perhaps sufficient to allow us to 
look beyond the obvious stylistic differences that sometimes 
appear to divide them and to accept, without troubling second 
thoughts, the identification of the two artists as one. 



Whomever one sees behind the unfolded biglietto lette- 
red IACO. BAR. VIGENNIS P. 1495, he knew Pacioli in 
1495, he apparently traveled to Urbino and knew something 
of Piero's art, and, judging from his style, he was a Venetian, 
working in a late Quattrocento tradition. In any event, it is 



Applications of the Art of Measurement 81 

Venice that draws artist and sitter together, and in the decade 
following 1495 everyone whom one would associate with Pa- 
cioli's portrait appears there. 

In Venice Pacioli began the peripatetic career that led 
him back and forth across Italy — to Rome, Naples, Urbino, 
Milan, Florence, Pisa, Bologna, Perugia, and elsewhere. In 
1464 he first carne to Venice, staying for almost a decade in 
the house of the merchant Antonio de Ropiansi on the Giu- 
decca as the tutor to the household's three sons 30 . During 
these years Pacioli followed the mathematics lessons of the 
Venetian Domenico Bragadino and others. And Venice was 
where he conversed about perspective with Giovanni and Gen- 
tile Bellini and with Hieronimo Malatini, another perspective 
teacher. In 1494, Pacioli returned to Venice to see his Stimma 
through the press, and he seems to have remained there into 
1495. And then, in 1508 and 1509 Pacioli was again in Ven- 
ice to deliver lectures on the fifth book of Euclid, and to see 
his by then completed Euclid translation (1509) and Divina 
proportione (1509) published. 

Even today Venice is small, and the world of artists and 
their admirers has never been very large. In these years Ja- 
copo de 'Barbari was in Venice, too, certainly from 1498 to 
1500 31 , when he was preparing the great panoramic map of 
Venice for the German merchant Anton Kolb, and was thus 
engaged on the vast perspective and surveying enterprise that 
took its participants into every nook and cranny, down every 
alley way, and up the stairwells of every belltower of the city. 

Dùrer's quest after the secrets of Italian art drew him to 
Italy twice, both times in these years and both times to Ven- 
ice. His first visit to Venice extended over 1494-1495, and 
his second from 1505-1507, the latter including his mysterious 



30 Staigmuller, « Biographische Skizze, » p. 87. 

31 G. Mazzariol, T. Pignatti, La pianta prospettica di Venezia del 
1500 disegnata da Jacopo de' Barbari (Venice 1962), p. 8. 



82 Chapter Four 

trip to Bologna to learn the secrets of perspective 32 . During 
his second trip Diirer writes back to his friend Pirckheimer, 
that although now « there are many better painters here than 
Master Jacob », Anton Kolb stili swears by him as the best 
painter in the world M . 

A few years earlier, in 1500, Barbari had been called to 
the court of Maximilian at Nuremberg, Dùrer's native city, 
in the capacity of portraitist and miniaturist 34 . Like Pacioli 
in Italy, Barbari became in Germany a spokesman for the 
inclusion of painting among the liberal arts. Pacioli had argued 
that just as music was part of the Quadrivium, so also should 
perspective be 35 — and thereby painting, too, would become 
a liberal art. Jacopo de 'Barbari pleads the same cause in a 
letter he addressed to Frederick the Wise, Elector of Saxony, 
around 1500. Painting, he wrote, should be declared the 
eighth liberal art, and hence put into the hands of noble men 
who would glorify it: « ...eser fata la otava arte liberale et 
eser levada de la mendicacione et eser posta in man de nobeli 
homeni, che la posi glorificare » 36 . Because of its sdentine 
foundations in arithmetic and geometry, painting belonged 
among the liberal arts — for there is no proportion without 
number, he wrote, and no form without geometry: 

E senza ese arte non poi essere pitura probabile fata da pitori, 
se non sarà periti nelle sopradite arte, prima nella giometria, 
po' aritmetica, lequal due necessita nelle commensuracione de 
proportione, che non poi essere proporcione senza numero, né 
poi essere forme senza giometria 37 . 



32 E. Panofsky, Mbrecht Diirer, 3rd ed. (Princeton 1948), I, pp. 8-9. 

33 Diirer, Schriftlicher Nachlass, ed. H. Rupprich, I (Berlin 1956), 44. 

34 L. Servolini, Jacopo de' Barbari (Padua 1944), p. 74, n. 20: « con- 
trafeter und illuminist. » 

35 Pacioli, Divina proportione, fol. 3r. 

36 P. Barocchi, Scritti d'arte del Cinquecento, I, (Milan 1971), 69. 

37 Ibid., pp. 66-67. 



Applications of the Art of Measurement 83 

Barbari's concern with perspective and proportion and his 
attempts to provide the art of painting with a theoretical and 
sdentine foundation were certainly reasons why Albrecht Dii- 
rer was interested in him. Possibly the two men met as early 
as 1495, during Dùrer's first stay in Venice. In any event, in 
a draft, written in 1523, of the dedication of his Proportions- 
lehre to Wilibald Pirckheimer, Durer describes his meeting 
with Jacopo, his introduction to the scientific aspeets of Bar- 
bari's art, and the catalytic effect Barbari had on his own 
theoretical studies. 

Dan ich selbs wolt lyber ein hochgelerten berumbten man jn 
solcher kunst hòrn und lesen, dan das ich als ein unbegriinter 
dofan schreiben soli. Jdoch so ich keinen find der do etwas 
beschriben hett van menschlicher mas zw machen dan einen man 
Jacobus genent, van Venedig geporn, ein libricher moler. Der 
wies mir man und weib, dy er aws der mas gemacht het, und 
das ich awff dyse tzeit liber sehen wolt was sein mainung 
wer gewest dan ein new kunigraich, und wen ichs hett, so 
wolt jch ims zw eren jn trug pringen, gemeinem nutz zw gut. 
Aber ich was zw der selben tzeit noch jung und het nie fan 
solchem ding gehòrt. Und dy kunst ward mir fast liben, und 
nam dy ding zw Sin, wy man solche ding mòcht zw wegen 
pringen. Dan mir wolt diser forgemelt Jacobus seinen grunt 
nit klerlich tzeigen das merckett ich woll an jm. Doch nam ich 
mein eygen ding fur mych, und las den Fitrufium, der beschreibt 
ein wenig van der glidmas eines mans. Also van oder aws den 
zweien obgenanten manen hab ich meinen anfang genumen, 
und hab dornoch aws meinem fur nemen gesucht van dag zw 
dag 38 . 



Although Diirer was unable to extract from Barbari his 
secrets — his canon of proportions and the principles on which 
it was based — he could turn to ancient and modem published 



38 Durer, Nachlass, I, 102. 



V 



84 Chapter Four 

sources, sudi as Vitruvius, Euclid, and Pacioli. Prompted by 
Barbari's example, aided by the humanist library of ancient 
works belonging to his friend Pirckheimer, and furnished with 
the printed treatises of modem Italian theoreticians, Dùrer 
composed his own treatises on measurement and proportion 
for German artists. 

In his projected manual for the education of the artist, 
Der Speis der Malerknaben, Diirer hoped to train aspiring 
painters in the foundation of art, measurement and proportion. 
The dedication of his Unterweisung der Messung (Nuremberg 
1525), the first part of the never completed Speis der Ma- 
lerknaben, spells out Dùrer's fundamental conviction that the 
art of measurement was the basis of painting. 

Die weyl aber die [Kunst der Messung] der recht grundt ist 
aller mallerey, hab ich mir fùrgenomen alien kùnstbegyrigen 
jungen, eyn anfang zùstellen, und ursach ziigeben damit sie sich 
der messunge zirckels und richtscheyt underwinden, unnd da- 
rauss die rechten warheyt erkennen unnd vor augen sehen mò- 
gen, damit sie nit alleyn zù kiinsten begirig werden, sonder 
auch zu eynem rechten und gròsseren verstant komen mogen 39 . 



The fourth and last book of the Unterweisung shows 
clear connections with the work of Dùrer's Italian « masters ». 
This book treats polyhedra, and Dùrer writes that these are 
of three kinds: first, the elongated and cylindrical (e.g., towers 
and columns); second, the pointed (e.g., pyramids); and third, 
the regular bodies, symmetrical in every respect ". 

Dùrer's definitions recali those of Piero's Libellus. Dùrer 



39 A. Durer, Underweysung der messung mit dem zirckel und richt- 
scheyt (1525; facsimile rpt. Dietikon-Zùrich 1966), verso of title page. 

40 Book IV, n. pag., see diagram 29. 



Applications of the Art of Measurement 85 

describes them as « Corpora die allenthalben gleych sind, von 
felderen, ecken und seyten, die der Euklides corpora regularia 
nennet, der beschreibt jr fùnffe, darumb das jr nit mer kùnnen 
sein, die in ein kugel darin sie allenthalben an rtiren verfasst 
miigen werdenn, die selben nach dem sie zìi vili dingen nutz 
sind, wil ich hie anzeygen » 41 . Piero, in his Libellus, had de- 
fined the « corpora »: « Multa sunt corpora lateribus consti- 
tuta quae in sperico corpore locari queunt, ita ut eorum anguli 
sperae superficiem omnes contingunt. Verum quinque ex eis 
tantummodo sunt regularia: hoc est, quae aequales bases ha- 
bent et latera » 42 . 

Piero had illustrated his Libellus and the Trattato d'abaco 
with fine pen drawings of the regular bodies in perspective, 
executed with a compass and ruler (figs. 17, 18, 19). Pacioli, 
when he published Piero's treatise in the Divina proportione, 
relied on large woodcuts depicting models of solid and hollow 
geometrical bodies, drawn in perspective (figs. 20, 21). Durer 
carried this process of illustration one step further. Instead of 
showing the regular and irregular bodies foreshortened, he 
provided two-dimensional diagrams as patterns for construc- 
ting from paper each of the five regular Euclidean bodies, nine 
of the irregular Archimedean bodies, and the sphere. With 
these patterns the construction of a set of geometrical bodies 
became a simple matter: « Aus disen dingen », he wrote, « mag 
man gar manicherley machen, so jr teyl aufeinander versetzt 
wirt, das zu dem aushauen der seulen und jren zirden di- 
net »" (àg. 21). 

A knowledge of these solids not only had useful applica- 
tion in architectural matters, but it had concrete practical 
advantages for painting. And so, Dùrer proceeds to demon- 



41 Ibid. 

42 Piero della Francesca, L'opera « De corporibus regularibus, » ed. 
G. Mancini (Rome 1916), p. 489. 

43 Durer, Underweysung, Bk. IV, n. pag., diagram 43. 



86 Chapter Four 

strate how to place them in a painting. « So ich daforen mani- 
cherley corporei wie man die mach anzeigt hab, wil ich auch 
leren so man soliche gemecht ansicht wie man die in ein gemei 
mtig pringen... » 44 . The example he chooses to illustrate is the 
cube (fig. 24), showing how to represent it in correct perspec- 
tive and how the six-faced regular body will cast shadows 
under specific light conditions » 45 . 

Following his experiments with light and shadow, Dùrer 
proceeds directly to the representation of the cube in perspec- 
tive giving two methods that rely on the intersection of the 
visual pyramid ". These belong to the sdentine demonstrations 
of « per spediva communis », i.e. mathematically worked out 
solutions rather than approximated ones. « In gleycher weyss 
wie ich den cubus in ein abgestolen gemei gebracht hab », 
Dùrer writes, « al so mag man alle cor por a die man in grund 
legen und aufziehen kan, durch sàliche weg in ein gemei prin- 
gen » 47 . This statement is cruciai to the development of his 
later Vroportionslehre, for his particular model of the cube 
can be generalized, and ali ordinary solid objects can be ren- 
dered in a geometrie manner. Dùrer then closes the Unter- 
weisung with artificial means of measurement for painters, the 
« per spediva artificialis » solutions, methods of seeing through 
a framed glass with one eye in a limited, fixed position, and 
for establishing the composition by means of threads running 
from the object to the wall in front of it, intersecting a small 
door. 

With his patterns for constructing regular and irregular 
solids and with his explicit diagrams for representing them 
foreshortened and under specific conditions of light, Dùrer 



44 Ibid., verso of diagram 51. 

45 Ibid., diagrams 52, 56, 57, 58. 

46 Ibid., diagrams 59, 60, 61, 62. 

47 Ibid., diagram 62. 



Applications of the Art of Measurement 87 

brought the theme of the Euclidean bodies into direct rela- 
tionship with the concerns of the practicing artist. The Italians, 
Piero and then Pacioli, had established the relationship of the 
regular bodies to the artist and architect on a theoretical basis, 
but the practical implications were less explicitly demonstra- 
ted. In the Trattato d'abaco the regular bodies were a compo- 
nent of the broader topic of solid geometry, but Piero's Abaco 
demonstrations were nonetheless dose to those of his later 
Libellus. These complex mathematical schemes of measurement 
showed the proportional relationship of regular bodies to a 
sphere and to one another, and thus established the theoretical 
basis for their construction. An artist who wanted to further 
the perspectival skills he acquired from the Prospectiva pin- 
gendi could sharpen them by following the mathematical dem- 
onstrations in the Libellus; in this way, too, his perspectival 
knowledge would be set on a firmer mathematical foundation. 
For Pacioli, the regular bodies took on a Platonic, and quasi- 
mystical significance. Thus their consideration, or contempla- 
tion, was an affair undertaken far above the ordinary level of 
everyday discourse. Because of the necessity of employing the 
golden section in constructing the dodecahedron — the Pla- 
tonic symbol of the Universe which subsumed the other four 
regular bodies — ali the regular solids were called « divine » 
and thought to have effetti ranging from dignissimo and ec- 
cellentissimo to inextimabile and mirabile 48 . Diirer in a sense 
brought the topic down to earth again by teaching aspiring 
artists in a practical way how to construct regular and irregu- 
lar bodies from paper and how to use them for exercises in 
perspective and in light and shade. His principal objective was 
to teach, through measurement, the correct representation of 
solid objects in space, and since ali objects could be treated 



48 Pacioli, Divina proportione, Part I, fols. 5v-7v. 



88 Chapter Four 

according to the model of the cube, the practical implications 
of the Euclidean bodies became infinitely more numerous * 9 . 

Not only could ali solid inanimate objects found in nature 
be rendered scientifically, and hence correctly, but in his Pro- 
portionslehre , Dùrer even treated anatomical renderings in 
geometrical terms. Book IV, dedicated to figurai movement, 
functions as a postscript to Dùrer's geometrical studies in the 
Vnterweisung der Messung 50 . First he shows how to foreshort- 
en faces by means of the « third outline » — a procedure he 
perhaps learned from Vincenzo Foppa's lost perspective manual 
(fig. 25) 51 , but one also closely related to Piero della France- 
sca's head and capital studies in the Prospectiva pingendi (figs. 



49 The relationship between the art of lettering and the study of 
measurement and proportion is made clear in Pacioli's directions in the 
Divina proportione for forming the letters of the alphabet. In his Unter- 
weisung der Messung Durer similarly shows how to measure and form 
the letters of the Latin as well as the Gothic {Fraktur) alphabet. His instruc- 
tions are intended for « builders and also painters, and those who need to 
place letters high up on a wall » (Bk. Ili, n. pag., see Latin letter A). A 
somewhat later pictorial source shows that humanist letter designers studied 
the regular bodies. Nicolaus Neufchatel's painting (1561) in the Alte 
Pinakothek, Munich, shows Johannes Neudorfer, the famous Nuremberg 
mathematician and calligrapher, teaching his son how to measure a hollow 
dodecahedron with a compass. Above them hangs a hollow cube, and around 
the frame of the canvas runs an inscription which is a masterpiece of lettering. 
Neudorfer was the first to write a set of artist biographies in the north, 
and in his work he attached a high value to the study of Euclid's Elements 
(Nachrichten von Kunstlern una Werkleuten [Vienna 1875], p. 48). For 
Neufchatel's painting see R. A. Peltzer, in Munchner Jahrbuch der bilden- 
den Kunst, III (1926), 192-94; Alte Pinakothek Munchen: Kurzes Ver- 
zeicbnis der Bilder (Munich 1957), p. 73, piate 118; Deutsche und nieder- 
làndische Malerei zwischen Renaissance und. Barock, Alte Pinakothek Mun- 
chen, Katalog I (Munich 1961), p. 36. A follower of Neufchatel painted 
Stephan Brechtel, also a mathematician and calligrapher, teaching his son 
the characteristics of an irregular polyhedron. Above them is suspended 
a hollow sphere (Peltzer, p. 195). 

50 A. Durer, Vier Bùcher von menschlicher Proportion (1528; facsi- 
mile rpt. Unterschneidheim 1969), also called the Proportionslehre. 

51 E. Panofsky, Dùrers Kunsttheorie (Berlin 1915), pp. 54-55. 



Applications of the Art of Measurement 89 

26, 27). From frontal and profile views it is shown how to 
find the « ground pian », as it were, or the third outline, of the 
whole head. And from a profile inscribed in a rectangle tipped 
upwards or downwards, Diirer demonstrated how to foreshort- 
en the head seen from above and below. Utilizing the « plans » 
of heads and whole bodies, the human figure may be ascribed 
the same properties as the inanimate cube, and thereby, in 
theory, it may be studied, outlined, and projected onto the 
picture surface as a regular body. 

Diirer continues, in the last book of the Proportionslehre, 
demonstrating the enclosure of limbs and other parts of the 
body within piane and solid geometrical figures (fig. 28). These 
figures, we recognize, are simplifications of the regular and 
irregular solids of the JJnterweisung, and consequendy the 
procedure Diirer outlines aids in understanding the volumes 
of the human body and its correct foreshortening. There exist 
drawing after drawing by Dùrer to chart his concern with 
geometricizing human figures, thus rendering them scientif- 
ically (figs. 29, 30) 52 . In their application to human figures 
the knowledge of abstract geometrical bodies is brought en- 
tirely into the realm of painting 53 . 



52 See esp. Das Skizzenbuch von Albrecbt Diirer in der Kònigl. Of- 
ferti. Bibliothek zu Dresden, ed. R. Bruck (Strassburg 1905). Also W. 
Strauss, The Complete Drawings of Albrecbt Dùrer (New York 1974), 
V, HP 1527/13; HP 1527/19; HP 1527/21; HP 1527/25 and passim. 
For Diirer's drawings showing the use of regular and irregular bodies in 
goldsmith designs, see Strauss, IV, 1526/12-1526/22. For polyhedron 
sketches, VI, AS 1514/19; AS 1514/20; and passim. In addition for Du- 
rer's drawings of human figures proportioned according to the « progressive 
proportion» see Strauss, V, HP: 1513/7; HP 1513/10; HP: 1513/14 
HP: 1513/16; HP: 1513/18. 

53 Such elaborations in the application of pure geometrical solids to 
the representation of the human figure are related to the small wooden 
models, with moveable limbs, that German artists drew. For the literature 
on these mannequins, see A. Weixlgartner, in Beitràge zur Kuntsgeschicb- 
te Franz Wickhoff gewidmet (Vienna 1903), pp. 80-90; J. v. Schlosser, 
in Jahrbuch der Kunstbistoriscben Sammlungen des Allerhóchsten Kaiser- 



90 Chapter Four 

In 1568, a book with a long, descriptive title appeared in 
Nuremberg, the goldsmith Wenzel Jamnitzer's Perspectiva 
corporum regularium, das ist ein fleyssige Fùrweisung wie die 
Fùnff Regulirten Córper, darvon Plato inn Timaeo, unnd 
Euclide s inn sein Elementis schreibt e te, durch einen son- 
derlichen, newen, behenden unnd gerechten Weg... gar kunst- 
lich inn die Perspectiva gebracht... werden mògen » M . Jamni- 
tzer's book represents the culmination of the researches of Pie- 
ro, Pacioli, and Diirer, for it illustrates in perspective, not 
only the five regular bodies, but also over one hundred and 
fifty other geometrical solids deriving from the five funda- 
mental forms (figs. 31). Owing to the great variety of geomet- 
rical bodies presented and to the high quality of Jost Amman's 
engravings, Jamnitzer's Perspectiva corporum regularium may 
deservedly be considered a virtuoso performance. Yet it rep- 
resents, in another sense, a simplification of the tradition 
Piero, Pacioli, and Diirer established. One indication of this 



hauses, XXXI (1913-14), 111-18; B. Degenhart, in Zeitschrift fur 
Kunstgeschichte, Vili (1939), 125-35; A. Weixlgartner, in Arstryck, 
Gòteborgs Kunstmuseum (1954), pp. 37-71. For Piero della Francesca 
and the use of models see Vasari-Milanesi, II, 498-99; J. v. Schlosser, 
« Kiinstlerprobleme der Friihrenaissance, II, Piero della Francesca, » 
Akademie der Wissenschaften in Wien, Philosophisch-historische Klasse, 
Sitzungsberichte, 214. Bd., 5 Abh. (1933), 10; E. Battisti, Piero della 
Francesca, I, 483, n. 187. From these Possen, or mannequins (which Durer 
specifically mentions in bis Proportionslehre), and from Durer's studies 
in the measurement of abstract geometrical bodies and their application to 
the representation of movement in the human figure, another type of ar- 
tists' manual was developed in the north. It is best exemplified by Erhard 
Schoen's Underweysung der Proportion und Stellung der Bossen ... (Nu- 
remberg 1538) and H. Lautensack's Des Cirkels und Richtscheyts ... 
(Frankfurt 1564). For these works see J. v. Schlosser, La letteratura ar- 
tistica (Florence 1964), pp. 278-80. 

54 Facsimile ed. Graz 1973. See also W. Jamnitzer, Orfèvre de la 
rigueur sensible: étude sur la « Perspectiva corporum regularium, » ed. A. 
Flocon (Paris 1964). For the relationship of regular bodies to goldsmith 
practices? K. Pechstein, « Jamnitzer-Studien, » Jahrbuch der Berliner Mu- 
seen, Vili (1966), 239. 



Applications of the Art of Measurement 91 

is that Jamnitzer fails to provide the reader with directions 
for costructing geometrical bodies. Moreover, little or 
nothing of the mathematical or proportional theory which 
lies behind the regular solids finds its way into the pages of 
Jamnitzer's work. Instead the geometrical forms have become 
academic stereometrie exercises for the study of perspective. 
Jamnitzer's work must be considered a kind of pattern book, 
or copybook, to be studied, perhaps, in conjunction with orai 
instruction. Nonetheless, one Italian theoretician, who knew 
the history of the subject, recognized the true origins of 
Jamnitzer's book. This theoretician was Egnazio Danti, who 
wrote: « Wenceslao Giannizzero Norinbergense, il quale ha 
messi in Prospettiva li corpi regolari, et altri composti, si come 
fece Pietro dal Borgo, se bene Fra Luca gli stampò sotto suo 
nome » 55 . 

Not long after the publication of Dùrer's Unterweisung der 
Messung and his Proportionslehre, each was translated from 
German into Latin, thus becoming more generally accessible 
to the flourishing community of Italian scholars and theore- 
ticians of the Cinquecento *. One of the most learned of these 
was Daniele Barbaro, Patriarch-Elect of Aquileia (1514-1570), 
Venetian ambassador to the Council of Trent, mathematician, 
astrologer, and inventor 57 . One of the most prolific and many- 
sided treatise writers of the sixteenth century, Barbaro re- 
veals, in his perspective works, a knowledgeable command of 
Piero's ideas. Barbaro's deep familiarity stemmed in part from 
a direct acquaintance with Piero's work, but his knowledge 



55 J. Barozzi Vignola, Le due regole della prospettiva pratica, ed. E. 
Danti (Rome 1583), Preface. 

56 Schlosser, Letteratura artistica, pp. 273-74. 

57 See G. Alberigo's entry in Dizionario biografico degli Italiani, VI, 
s.v. « Barbaro, Daniele »; Andrea Palladio 1508-1580, ed. H. Burns, Arts 
Council Exhibition (London 1975), pp. 94-96. 



92 Chapter Four 

of Piero derived also in part from the writings of Pacioli and 
those of Dùrer. Barbaro himself wrote two perspective man- 
uals. Their titles are practically identical, though one was 
published and the other remained in manuscript. The first, 
La pratica della perspettiva, was published in Venice in 1569. 
It contains nine chapters on various aspects of perspective 
science, with frequent references to Piero della Francesca. The 
title of Barbaro's second, «La pratica della prospettiva,» varies 
by a single letter. Nonetheless it is a wholly separate treatise. 
It exists in manuscript in the Biblioteca Marciana in Venice, 
and it is precisely a treatise on the regular and irregular 
bodies 58 . 

Barbaro's marked interest in perspective and regular bodies, 
as he himself relates, initially grew out of his Vitruvian studies. 
He, like Vitruvius, refers to perspective as « scenogra- 
phia » 59 . His published treatise on perspective, La pratica 
della perspettiva, leads from detailed instructions concerning 
polygons and polyhedra to the correct painted representations 
of buildings and their architectural parts to the end of con- 
structing theatrical scenery. Chapters I and II of the Perspet- 
tiva teach the principles of perspective and the geometrical 
construction and foreshortening of polygons; Chapter III, by 
far the longest, the « modo di levare i corpi dalle piante », 
e.g., the way to construct the three-dimensional dodecahedron 
from its two-dimensional pian, the pentagon; Chapter IV con- 
cerns « sceno graphia ». 

Barbaro's discussion of the regular and irregular solids 
in Chapter III of the Perspettiva is far more exhaustive than 



58 D. Barbaro, La pratica della perspettiva (Venice 1569), and idem, 
« La pratica della prospettiva, » Venice, Bibl. Marciana, MS. It. IV, 39 
( = 5446). The ms. of the published book is also in the Marciana, MS. It. 
IV, 40 (=5447). 

59 Barbaro, Perspettiva (1569), p. 3; idem.* «Prospettiva,» MS., 
fol. 298r; Vitruvius, De architectura (VII, Preface, 11). 



Applications of the Art of Measurement 93 

anything we have seen before in Piero, in Pacioli, or even 
in Diirer. It is Barbaro who, besides giving patterns for the 
bodies and instructions in shading, gives explicit and very 
detailed directions for rendering most of the regular and ir- 
regular bodies in perspective with a compass and mie. His 
instructions for constructing the dodecahedron illustrate well 
the detailed character of his treatment. 

Io ponerò qui appresso quello, che appartiene alla pratica. Et 
perche tutti questi corpi regulari sono circonscritti dalla sphera, 
cioè con tutti gli anguli loro toccarebbero la concavità d'una 
sphera nella quale fussero rinchiusi, però nella formazione delle 
loro piante perfette, si formano in uno circolo. Facciasi adunque 
sopra '1 centro a uno circolo, & sia partito in dieci parti equali 
b, e, d, e, f, g, h, i, k, 1, e alternamente sopra quelle parti fac- 
ciansi due soperficie di cinque lati equali, l'una sia bdfhk, l'al- 
tro cegil, etc (fig. 32) 60 . 

Drawing regular and irregular bodies in perspective was 
evidently, for Barbaro, an exercise in learning how to render 
buildings. In stage design, that is, in « scenographia », there 
is but a single vanishing point for the painted architecture, 
and the individuai components must be established with ref- 
erence to this point. Starting with the relatively simple reg- 
ular bodies, one proceeds to complicated irregular solids, and 
then ultimately to an accurate, correct perspectival rendering 
of columns, bases, capitals, architraves, portals, vaults, and 
other architectural components (fig. 33). Moreover, Barba- 
ro's demonstrations show how to shade these elements. Thus 
the study of abstract polygons and polyhedra was, in Barba- 



60 Barbaro, Perspectiva (1569), pp. 49-50. Cf. Durer, Underweysung, 
Bk. IV, diagram 33, where the overlapping pentagons are shown but where 
no verbal instructions are given. Danti's commentary to Vignola's Due 
regole, p. 143, mentions the lack of instruction in Jamnitzer's treatise: 
« se bene ha delineate solamente le figure senza scrivervi attorno cosa nes- 
suna. » One may imagine Barbaro's directions were welcome aids. 



94 Chapter Four 

ro's published Pratica della perspettiva, intended as training 
in the correct representation of objects found in nature, and 
in this way the usefulness of understanding regular and irreg- 
ular bodies is brought to bear on both painting and architec- 
ture. 

Barbaro's Venetian manuscript, « La pratica della prospet- 
tiva », directly relates the study of geometrical bodies to per- 
spective, and hence to the practical concerns of artists: « con- 
viene che bora dimostrarne» alcuna delle forme irregolari, perche 
oltra che elle sono di maggior numero et molto più pertinenti 
al dipintore che le regolare » 61 . Entitled « Prospettiva », Bar- 
baro's treatise contains a concise introduction to the principles 
of perspective, but primarily, the treatise is dedicated to the 
regular and irregular bodies. 

The knowledge of perspective, Barbaro writes, is a prere- 
quisite to understanding the treatise at hand. In a treatise not 
for beginners, he recapitulates the principles of perspective ex- 
plained at length in his published book. Barbaro loses little time 
arriving at the subject which truly interests him, the construc- 
tion and perspectival rendering of geometrical bodies. As in 
his published book on perspective, here, too, Barbaro expli- 
citly conneets the ability to render these stereometrie solids 
with skills in rendering architectural motifs. 

Hor perche gli e necessario volendo per arti di prospettiva di- 
signari sopra uno quadro qualche architettura, over altra cosa 
che stai debbia, come essa perspettiva richiede, intendo dir 
il suo perfetto, non dico perfetto quanto a tutta la forma: per- 
che presupono che uno dipintor, oltre la perspettiva debba 
perfettamente intendere quali siano le buone forme: si di gli 
edifici]', come da tutte le altre cose: volendo nel arte sua proce- 
dere secondo che ella richiede: ma solamente intendo per il 
suo perfetto haver cognitione che forma, che li habbia, ciò è 
triangolare, quadrata, pentagona, over di altra sorte: et come 



61 Barbaro, « Prospettiva », » MS., fol. 19r. 



Applications of the Art of Measurement 95 

tutti li anguli di essa architettura, over altra cosa essendo ma- 
terialmente costruita 62 . 

On folio 12r Barbaro introduces the ways of foreshorten- 
ing polygons, and, on folio 25r he continues with instructions 
in constructing,in shading, and in rendering perspectivally the 
regular and irregular solids. The figures are not always drawn 
in, but their designs are incised by the compass points into 
the paper, and hence ready to be inked in. Patterns for an 
immense variety of geometrical solids are given, and examples 
of geometrical solids inscribed in others are shown. From the 
more or less simple geometrical bodies, there are developed 
elevated and stellated regular bodies and intricate mazzocchi 
(figs. 34-36). A whole section, extending from folio 168r 
through 215r, is devoted to the sphere, and this theme, too, 
is brought specifically into the painter's domain: « ...et se un 
pittore volesse far in pittura un corpo spherico, over balla, 
tirra col compasso un circolo, et quello date le sue ombre di- 
ligentemente havrà fatto lo detto corpo spherico » 63 . 

Most of the treatise, approximately 270 folios of a total 
of 310, is devoted to the correct perspectival drawing of geo- 
metrical bodies. Barbaro admits the difficulty of rendering the 



62 Ibid., fol. 7v. 

63 Ibid., fol. 168r. A related exercise in Piero's Prospettiva pingendi 
(p. 210) seems to have been overlooked in ali the literature on the egg 
in the Brera panel (most recently: I. Ragusa, « The Egg Reopened, » Art 
Bulletin, LUI [1971], 435-43): « Acade al le volte de volere dimostrare 
sopra de alcuna taula o spazzo, o socio a sularo, alcuno corpo o sopra o 
sodo a quelli posto, siccomo sopra delli spacci tu volesse circulare et con- 
torneare corpi che paressero elevati, cioè casse, deschi, palle, animali et 
similmente sopra taula da mangiare vasi, candelieri e altri corpi, così sodo 
sulari o socto volte anelli o altre cose che pendessero, che ad termine pares- 
sero commo veri. » Piero was interested in the correct representation of 
solids suspended from a vault. Although he, like Barbaro, was concerned 
mainly with circles and spheres, there are instructions for representing the 
ovoid form in Durer {Underweysung, Bk. I, diagrams 22, 34). Stili another 
egg in F. Benaglio's Sacra conversazione, S. Bernardino, Verona, and in 
Ercole de' Roberti's S. Girolamo, S. Petronio, Bologna. 



96 Chapter Four 

most complicateci of these forms, but he nevertheless urges 
their study, even if the student must read his book « quattro, 
et sei volte con grande difficoltà et non senza gran fatica » M . 
The work, in fact, bespeaks a fascination with these stereo- 
metrie solids similar to that of Luca Pacioli. But unlike Pacio- 
li, Barbaro's discussions are without philosophical or meta- 
physical speculations. Instead, for the Venetian, the study of 
regular and irregular polyhedra afforded artistic exercises in 
developing a hand practiced in the accurate, yet easy rendition 
of three-dimensional space and stereometrical volume. 

•k -k Ve 

The measurement of regular and irregular bodies, which 
Piero initiated, became in time a standard feature of the 
perspective manuals used by artists and architeets 65 . This 
stereometrical mensuration was dependent upon the fixed 
ratios existing among the Euclidean solids and the sphere. 
And, it was a knowledge of these invariable proportions that 
permitted the accurate measurement of abstract polygons and 
polyhedra. 

In the foregoing remarks concerning the legacy of Piero 
in the work of Pacioli, Dùrer, and Barbaro, we have concen- 
trated on the relationship of the regular bodies to perspectiv- 
al study. Perspective was emphasized in order to underline 



64 Barbaro, « Prospettiva, » MS., fol. 252v. 

65 Besides those noted, see also: Cosimo Bartoli, Del modo di mi- 
surare (Venice 1564), fols. 83-91; Jean Francois Niceron, La perspective 
curieuse, ou Magie artificielle (Paris 1638), Book I, pp. 28-36, and illustra- 
tions; Giovanni Amico, L'architetto prattico (Palermo 1726-50), II, 141, 
fìgs. 21, 22; Elementi di perspettiva secondo li principii di Brook Taylor 
con varie aggiunte ... del Padre Francesco ]acquier (Rome 1755), pp. 46-47, 
fìgs. IX, X; Bernardo Antonio Vittone, Istruzioni elementari per indirizzo 
de' giovani allo studio dell'architettura civile (Lugano 1760), pp. 29-30; 
Eustachio Zanotti, Trattato teorico-pratico di prospettiva (Bologna 1766), 
p. 75 ff., fìgs 36-42; Ferdinando Galli Bibiena, Direzioni a' giovani stu- 
denti nel disegno dell'architettura civile (Bologna 1777), Preface, pp. 30- 
36, and illustrations. 



Applications of the Art of Measurement 97 

the relationship between Piero's Abaco and Libellus and his 
perspective treatise. But the regular bodies have many faces, 
and they exercised yet another fascination over Piero's fol- 
lower, Luca Pacioli, whose enchantment with proportion is 
epitomized in the title of his treatise « divina proportione ». 
Thus proportion, rather than perspective, could have been 
our theme. Within the realm of the regular bodies, perspective 
and proportion are often but two sides of the same concetti. 
Much of what we discover in reading Piero's mathematical 
works is just this measuring of regular bodies, set one within 
the other and within a sphere, through proportional rela- 
tionships derived from Euclid's theorems. In turn, Piero ap- 
plied numbers to these proportional relationships, thereby 
deriving a system of proportional measurement. 

Along these lines, Pacioli's statements concerning pro- 
portion lay bare the underlying congruity of Piero's three 
treatises. Proportion, writes Pacioli, is the mother and queen 
of ali the arts — none of them can be practiced without it 
(« Tu troverai la proportione de tutte esser madre e regina: 
e senza lei niuna poter se exercitare »). As a part of perspective, 
proportion will determine the size of a human figure both in 
relation to its setting and in terms of its distance from the 
viewer. And proportion, too, will determine where a figure 
should be placed in the scheme of the entire painting tó . AH 
this, Pacioli continues, Piero has demonstrated in his Pro- 
spettiva pingendi where « de le 10 parole le 9 recercano la 
proportione ». 



66 Pacioli, Summa, fol. 68v. «Questo el prova prospettiva in sue 
picture. Le quali se ala statura de una figura humana: non li da la sua 
debita grossecga negli occhi de chi la guarda: mai ben responde ... E cosi 
nelli piani dove hano apostare tal figura: molti li convene haver cura: de 
farla stare con debita proportione de distantia ... E cosi facendo una figura 
a sedere sotto qualche fornice: bisogna che la proportionino in tal modo: 
che levandose in piedi: ella non desse del capo fore coperchio. E cosi in 
altri liniamenti e dispositioni de qualunche altra figura si fosse. 



Appendix I 
Luca Pacioli and the «Trattato d'abaco» 



Piero della Francesca 's mathematical pursuits constitute an 
integrai part of his biography by Vasari. From his youth, Va- 
sari relates, Piero applied himself to the mathematical sciences, 
and even after he turned to painting in his fifteenth year he 
never ceased studying them — in fact, he made « maraviglioso 
frutto ed in quelle e nella pittura ». Learned in perspective 
and highly knowledgeable concerning the secrets of Euclid, Pie- 
ro understood the properties of geometrical bodies better than 
any master of his time. The most useful elucidations of these 
matters which we possess, Vasari tells us, are from his hand. 
Owing to his writings, Piero merits being called the « miglior 
geometra che fusse nei tempi suoi » 1 . 

The contributions of Piero matematico went unrecognized, 
Vasari continues, because, after Piero's death, the fruits 
of his labors were published under the name of his compatriot, 
Fra Luca Pacioli of Borgo San Sepolcro: « e venuto Piero in 
vecchiezza ed a morte, dopo avere scritto molti libri, maestro 
Luca detto, usurpandoli per se stesso, gli fece stampare come 
suoi, essendogli pervenuti quelli alle mani dopo la morte del 



Vasari-Milanesi, II, 487-501. 



Luca Pacioli and the « Trattato d'abaco » 99 

maestro » 2 . Vasari neglected, however, to support his con- 
tention by specifying which o£ Piero's books Pacioli had 
usurped. Sirice, with the passage of time and the multiplica- 
tion of printed works treating geometry, perspective, and the 
regular bodies, most of Piero's manuscript writings were lost 
sight of, Vasari's allegation of plagiarism long persisted as an 
issue, mainly in commentaries on Vasari's vita of Piero. None- 
theless, there was little concrete evidence to bring to bear 
upon the question. 

Then, in 1880, the German scholar Max Jordan uncov- 
ered Piero's Libellus de quinque corporibus regularibus among 
the Urbino manuscripts in the Vatican Library (Vat. Urb. lat. 
632). By comparing Piero's Vatican Libellus with Luca Pa- 
cioli's « Libellus in tres partiales tractatus divisus quinque 
corporum regularium et dependentium », a treatise dedicated 
to Piero Soderini, appended to Pacioli's Divina proportione, 
and published in Venice in 1509, Jordan recognized that Pa- 
cioli's « Libellus » on the five regular bodies was only a di- 
rect Italian translation of Piero's Latin Libellus in the Va- 
tican 3 . Thereby Jordan was able to confirm that Vasari's al- 
legation of Pacioli's plagiarism was substantially correct. 



2 Ibid., p. 498. Precisely at the time Vasari was preparing the first 
edition of his Lives, in 1546, a clamorous case of plagiarism carne to the 
fore in Florence. Perhaps significantly, the issue at hand was a mathema- 
tical one, the solution of the cubie equation (see above, Chapter II, p. 23 f.). 
Its two main protagonists, Niccolò Tartaglia and Gerolamo Cardano, 
marked their dispute indelibly in their longer publications and with a flurry 
of short polemical traets that spread throughout Italy. It seems highly pos- 
sible that this controversy played a róle in Vasari's formulation of Piero's 
vita. 

3 M. Jordan, « Der vermisste Traktat des Piero della Francesca 
ùber die fùnf regelmassigen Korper, » Jahrbuch der Kòniglich Preussi- 
schen Kunstsammlungen, I (1880), 112-18. 

For the question of Pacioli's plagiarism and the literature leading up 
to the discovery of the Libellus, see the editions of the Lives edited by 
G. Della Valle (III, 1971, 248n); G. Antonelli (V, 1828, 14n); G. 
Maselli (1832-38, p. 297, n. 2); L. Schorn (V, 1837, 297, n. 4); F. Ra- 



100 Appendix One 

It has not been noted, however, that Pacioli's treatise on 
the measurement of the five regular bodies, appended to the 
Divina prò por t ione in 1509, was preceded by a smaller trea- 
tise of some eleven pages on the same subject. Entitled « Par- 
ticularis tractatus circa corpora regularia et ordinaria », and 
dedicated to Guidobaldo da Montefeltro, Duke of Urbino, it 
consti tutes the final chapter of Pacioli's Stimma de arithme fi- 
ca, geometria, proportioni et proportionalità, published in Ven- 
ice in 1494 4 . Pacioli's introduction to the work is informa- 
tive and bespeaks his own genuine fascination with the sub- 
ject. 

E benché di sopra in questo nella distintion .6 a . al capitolo .4°. 
de la mesura de la spera succintamente fosse ditto abastanca: 
non dimeno me par qui excelso Duca particularmente dire de 
alquanti corpi essentiali in ditta spera locabili de li quali unan- 
golo toccando subito tutti toccano: e principalmente lo fo per 



nalli (II, 1848, 791, n. 2); Società di amatori delle arti belle (IV, 1848, 
14, n. 4); G. Milanesi (III, 1878, 487, n. 2). Also: separate ed. of the 
vita of Piero, with notes by F. Gherardi Dragomanni (Florence 1835), 
p. 19, n. 1; G. Bossi, Del Cenacolo di Leonardo da Vinci (Milan 1810), 
p. 10; G. Tiraboschi, Storia della letteratura italiana, VII, part 2 (Milan 
1824), 751; L. Pungileoni, in Giornale arcadico di scienze ed arti, LXII 
{1835), 214ff; LXIV (1837), 186ff; G. Gaye, in Kunstblatt, XVII (1836), 
no. 69, 284-7; E. Harzen, in Naumann's Archiv fiir zeichende Kùnste, II 
(1856), 237; H. Janitschek, in Kunstchronik, XIII (1879), 670-74. Un- 
known to Jordan, the manuscript of the Libellus had already been identi- 
fied by J. Dennistoun, around 1850, in the Vatican library. See his Me- 
moirs of the Dukes of Urbino, 2nd ed. (London 1909), II, 205-06. The 
Trattato d'abaco may well have been the manuscript by Piero that 
disappeared from the library of the Franceschi Marini family, Piero's 
descendants, around 1835 and that found its way into G. Libri's posses- 
sion (above, Chapter II, p. 22). See Vasari, Cinque vite annotate, ed. G. 
Mancini (Florence 1917), p. 210, n. 2. A dose reading of the commen- 
taries to Vasari's vita of Piero shows that around 1832 (Maselli, p. 297, 
n. 2) there were manuscripts by Piero in the possession of the France- 
schi Marini in Sansepolcro. By 1848, however (Società di amatori delle 
arti belle, IV, 13 n. 2), they were no longer there. 
4 Summa arithmetica, Part II, fols. 68v-73v. 



Luca Pacioli and the « Trattato d'abaco » 101 

la notitia de li .5. regulari di quali Euclide a pieno nelli ultimi 
soi libri scientificamente tratta. Di che me pare non inutile a 
ponere de loro certi casi acio li pratici vulgari ancora essi qual- 
che dolceeca di loro dimensioni sentine Di questi fra philo- 
sophy si fa gran discussioni. E maxime se bene el thymeo del 
Divin philosopho Platone (secondo lo Aurelio doctor sancto 
Augustin) con diligentia f atende. Dove de universi natura dif- 
fusamente parlando spesso a suo proposito li induci. Attribuen- 
dolo lor forme separatamente ali .5. corpi semplici: cioè. Terra. 
Aqua. Aeri. Fuoco. E Cielo... Siche di questi le sequenti peti- 
tioni inquiriranno el loro modo operativo abastanca pratica in- 
segnaremo commo apresto se intenderà 5 . 



The « corpora regularia et ordinaria » of the title are the 
five regular bodies. Succinctly Pacioli defines them as just 
those bodies that, if placed in a sphere and if one of the ver- 
tices touches the surface of the sphere, then ali the vertices 
will do so. He also states that they are fully treated by Euclid 
in the last books of the Elements, and he recalls the Platonic 
significance of the five regular bodies as attributes of earth, 
water, air, fire, and the heavens. The fifty-six exercises that 
follow Pacioli's introduction treat, for the most part, tetrahe- 
dra, cubes, irregular eight and fourteen-faced solids, dodeca- 
hedra, icosahedra, octahedra, and the sphere. These solid geom- 
etry lessons in Pacioli's « Particularis tractatus » were taken 
from Piero della Francesca's Trattato d'abaco, folios 105r 
through 120r, and they are the stereometrical exercises we 
discussed above in Chapter II. Over half of the problems were 
reproduced with no change from Piero's work, some were 
made more concise, others considerably shortened. A few were 
lengthened for further clarification and a few totally reworked 6 . 
Ali in ali, what we find at the end of Pacioli's Summa 



5 Ibid., fol. 68v. 

6 Twenty-seven of Pacioli's exercises are reproduced from the Trat- 
tato d'abaco almost word for word. They are nos. 3-5, 11, 14, 16, 19, 20, 
25, 27-30, 32, 33, 35, 37-44, 46, 54, 55. Ten were made more concise by 



102 Appendix One 

arithmetica represents Piero's preliminary work towards the 
Libellus de quinque corporibus regularibus, the solid geometry 
lessons of his Trattato d'abaco \ 



shortening sentences, or minor changes in wording — nos, 6, 9, 10, 13, 15, 
17, 21, 26, 34, 48. Six were considerably abbreviated — nos. 2, 7, 8, 22, 
52, 56; three lengthened for clarification — nos. 12, 45, 47; four partially 
reworked — 18, 51, 53, and the second problem numbered 53. Only five 
were completely revised, nos. 23, 24, 36, 49, 50. See Table I for the 
corresponding exercise numbers in the Summa arithmetica and the Trat- 
tato d'abaco. 

7 Exercise 31 in the Summa has no corresponding exercise in the Abaco. 
It does not, however, represent an addition by Pacioli because it appears 
later in Piero's Libellus (II, 37). This circumstance suggests that Pacioli 
had before him, not the Laurenziana ms. of the Abaco, but rather a faith- 
ful version of it. On the other hand exercise no. 4 on fol. 114r of Piero's 
Abaco does not appear at ali in Pacioli's Summa. Nonetheless, a know- 
ledge of its solution was necessary to follow the solution to Summa no. 
35. Its omission, by Pacioli or the printer, was undoubtedly uninten- 
tional. 

Inasmuch as Pacioli worked from a manuscript version of the Lau- 
renziana Abaco, not the Laurenziana Abaco (Ashb. 280) itself, does the 
« Particularis tractatus » of Pacioli's Summa arithmetica reproduce this 
lost manuscript of Piero's Abaco faithfully, as was the case with Piero's 
Libellus printed in Pacioli's Divina proportione? In other words, who 
was responsible for the divergences of the « Particularis tractatus » from 
the Abaco, as we have it in the Laurenziana version? Piero, in an interme- 
diate manuscript devoted to the regular bodies, or Pacioli himself? Pa- 
cioli's responsibility for the changes is indicated by a dose comparison 
of the 38 problems common to the Summa, the Abaco, and the Libellus 
(compare Tables I and II). Were Piero responsible for the variations in- 
troduced in the regular bodies treatise of the Summa, some movement 
towards the later Libellus should be discernible in these modifications. 
This is not the case. The exercises in the Summa that differ most markedly 
from their predecessors in the Abaco are nos. 23, 24, 36, 49, and 50. These 
problems relate no more closely to the corresponding exercises in the Li- 
bellus than they do to those in the Abaco. Thus the authorship of the 
variations between the Abaco and the « Particularis tractatus » must be as- 
signed principally to Luca Pacioli. This conclusion is confirmed by the 
character of some of the minor variations in wording, clarity, and numerical 
simplification, which may also be assigned to Pacioli. Furthermore, in some 
of the variant passages Pacioli actually refers back to earlier parts of the 
Summa proper, for instance, no. 23: « te mostrai ala terga di [stintion] nel 
.6." capitolo, » and in the same problem: « commo to mostro al suo luogo 
nella arithmetica. » 



Luca Pacioli and the « Trattato d'abaco » 103 

A representative example of Pacioli's use of Piero's work 
is exercise 29 of the Summa, where it is demonstrated how 
to find the diameter of a sphere into which an icosahedron, 
whose area is 200 braccia, has been inscribed. We read in the 

Summa: 

Eglie un ,20. base triangulari equilatero la cui superficie e brac- 
cia .200. domando del diametro de la spera dove sia descripto. 
Tu hai per la precedente che el .20. base che la sua superficie 
sia .200. che il suo lato e .R R. 533 1/3. e per la 2.a del .20. 
base hai che illato che e .4. de diametro .40. per. R.320. E per- 
che tu hai illato che .R R. reca .4.a. .R R. fa .256. e reca .40. 
a .R. fa .1600. e poi reca a .R.320. fa .102400. e hai .1600. 
per .R.102400. Ora di cosi se .256. de lato da de diametro 
.1600. per .R.102400. che darà .533 1/3. Montiplica 533 1/3 
via .1600. fa .853333 1/3. parti per .256. nevene .3333 1/3. 
ora reca a .R.533 1/3. fa 284444 4/9. montiplicalo con .102400. 
farà .29127311111 1/9. el quale parti per .256. recato a .R. fa 
.65536. nevene .429188 26068/32489. e hai 3333 1/3 per 
.R. 429188 26068/32489 adonca di chel diametro de la spera 
dove e discripto ditto corpo base .20. triangolari che la sua su- 
perficia sia .200. sia la .R. de la summa che fa la .R R. 429188 
26068/32489. posta sopra la .R. 3333 1/3 ecc. 



This is clearly taken from the Trattato d'abaco, fol. 
113r (3). 

Egl'è uno 20-baxe triangulare equilatero, che la superficie sua 
è 200 bracci. Domando del diametro de la spera dov'è descricto. 
Ai, per la precedente, che il 20-baxe che la superficie sua è 
200, che il suo lato è radici de la radici de 533 1/3. Et per 
la seconda de' 20-baxe ài che il lato ch'è 4 dà de diametro 40 
più radici de 320. Et perché tu ài il lato ch'è radici de radici, 
reca 4 a radici de radici fa 256, et reca 40 a radici fa 1600 et 
reca 320 a radici fa 102400, et ài 1600 più radici 102400. Hora 
di' così: se 256 de lato dà de diametro 1600 più radici de 
102400, che darà 533 1/3? Montiplica 533 1/3 via 1600 fa 
853333 1/3, partilo per 256 ne vene 3333 1/3. Hora reca a 
radici 533 1/3 fa 284444 4/9, montiplicalo con 102400 fa 
29127311111 1/9, il quale parti per 256 recato a radici, ch'è 



104 Appendix One 

65536, ne vene 429188 26068/32489. Et ài 3333 1/33 più ra- 
dici de 429188 26068/32489. Adunqua di' che il diametro de 
la spera, dove è descricto il corpo de 20 baxe triangulare che la 
sua superficie è 200, sia radici de la somma che fa la radici de 
la radici de 429188 26068/32489 posta sopra la radici de 3333 
1/3. 



Absolutely conclusive instances of Pacioli's use of Piero's 
work in the Summa are round in exercises 19 and 20. These 
represent two exceptions in Piero's Abaco, the two problems 
in measurement which do not concern abstract polygons and 
polyhedra but rather everyday objects. They treat the deter- 
mination of the « quadratura » of a barrel and that of a statue. 
Problems in measuring barrels are frequently encountered 
in abbaco handbooks, but the measurement of a statue is ap- 
parently unusual. (In the method of solution, however, it makes 
little difference what kind of a solid one measures.) Here is 
Pacioli's solution; the changes he introduces in Piero's text 
are italicized. 

Per che ale volte acade a mesurare corpi inregolari quali non 
se posano mesurare per linee, commo statue de marmo o de 
metallo cioè figure de animali rationali o inrationali dico che 
tenga questo modo a quadrarli apresso quello che di sopra su- 
cintamente dissi in la domanda .43. verbigratias. Metiamo che 
tu voglia sapere questo e quadrata una statua de homo nuda 
che sia .3. braccia de longhecca proportionata. Dico che tu fac- 
cia un vaso de legno longo braccia .3 1/4. e largo .1 1/2. e 
alto .1. el quale sia quadro: cioè con angoli retti e sia bene 
stagno siche laqua non desca ponto: poi lo metti bene piano 
alivello e metti dentro tantaqua che agionga a un terco alorlo 
di sopra. Poi fa un segno dove agiongni laqua e mettivi dentro 
la statua che tu voi mesurare e vedi quanto e cresciuta laqua 
e fa unaltro segno a sommo laqua ritto a quello de prima. Poi 
mesura dal primo segno al secondo e vedi quanto eglie: met- 
tiamo che sia 1/4 ora montiplica la longhecca del vaso che .3 
1/4. con la larghecca che .1 1/2. farà 4 7/8. el quale montipli- 
cata per 1/4 che creve laqua fa .1 7/32. e tanto e quadrata 



Luca Pacioli and the « Trattato d'abaco » 105 

ditta statua: e così obsua amesurare tali corpi se fosse ben un 
par de buoi con un carro de fieno ecc. 



Pacioli referred to his inclusion of the « Particularis trac- 
tatus circa corpora regularia et ordinaria » in the Stimma 
arithmetica in Chapter 69 of his Divina proportione, which 
dates from 1497 8 . The measurement of the five regular bodies, 
he wrote, would not be treated in the Divina proportione be- 
cause he had already composed a treatise on their measurement, 
dedicated to Guidobaldo da Montefeltro which was easy 
enough to come by. 

Segue a doverse dire dela dimensione deli corpi regulari e de 
loro dependenti. Onde de dicti regulari non mi curo altramente 
qui extenderme per haverne già composto particular tractato 
alo illustrissimo affine da vostra Ducale celsitudine Guido 
Ubaldo Duca de Urbino nella nostra opera a Sua Signoria dicata. 
e al lectore facile a quella sia el recorso per essere ala comune 
utilità pervenuta commo dinanze fo detto. E in questa vostra 
inclita cita asai se ne trovano. La cui mesura tanto e più spe- 
culativa quanto più degli altri corpi sonno quelli più excellenti 
e perfecti 9 . 



Pacioli writes « per haverne già composto particular trac- 
tato », and his introduction to the « Particularis tractatus » 
itself offers no indication of another author. Thus there was 
no reason to believe Pacioli was not its author. In the Summa 
arithmetica, then, there is a further substantiation of Vasari's 
contention that Fra Luca Pacioli published the works of « quel 
gran vecchio » under his own name. One further conclusion 
may be drawn concerning Pacioli's 1509 publication of the 
Libellus de quinque corporibus regularibus. The reader of the 



8 L. Pacioli, Divina proportione (Venice 1509), colophon on fol. 23r. 

9 Ibid., fol. 21v. 



106 Appendix One 

manuscript version of Pacioli's Divina proportione (1497) who 
wanted to learn about the measurement of regular bodies is 
referred to the earlier treatise on that subject published in Pa- 
cioli's Summa de arithmetica, geometria, proportioni et pro- 
portionalità rather than to the Libellus, immediately at hand in 
the Divina proportione. Thus it must have been after the com- 
pletion of his manuscript and toward the time of publication 
of the book that Pacioli decided to translate and publish the 
Libellus, which he appended to the printed edition of the Di- 
vina proportione of 1509. 



Appendix II 



Piero's Manuscripts and Pacioli's Adaptations: 
Textual Questions 



As we have seen, Luca Pacioli's work on the five regular 
bodies was largely derived from Piero della Francesca's. Both 
Fra Luca's « Particularis tractatus circa corpora regularia et 
ordinaria, » contained in his Summa de arithmetica, geometria, 
proportioni et proportionalità of 1494, and his « Libellus in 
tres partiales tractatus divisus quinque corporum regularium et 
dependentium », printed in his Divina proportione of 1509, 
represent little more than versions of two of Piero's treatises. 
Piero's Trattato d'abaco (Bibl. Laurenziana, Ashb. 280) pro- 
vides the basis for the regular body treatise Pacioli printed in 
his Summa arithmetica, and Piero's Libellus de quinque cor- 
poribus regularibus (Vat.Urb.lat. 632), the basis for the reg- 
ular body treatise Pacioli subsequently published in his Di- 
vina proportione. 

This doublé set of parallel circumstances raises several 
questions whose answers offer some insight into the character 
of Piero's originai manuscripts. 

Did Piero compose the Libellus in Latin, as sometimes sug- 
gested *, or did he write it in Italian and have it translated to 



1 For instance, L. Olschki, Geschichte der neusprachlichen wissen- 
schaftlichen Literatur (1919; rpt. Vaduz 1965), I, 216, n. 2. 



108 Appendix Two 

present to the Duke of Urbino? If Piero's Latin Libellus rep- 
resents, then, the work of a translator, is Pacioli's Italian 
version of it, contained in the Divina proportione, a successive 
translation into Italian, or is it simply the publication of Pie- 
ro's originai Italian manuscript? And finally, to what extent, 
if at ali, did Pacioli revise Piero's material before he published 
it? 

The answers to these questions arise from a dose compar- 
ison of the four texts in question. To anticipate conclusions, 
it seems clear that Piero wrote the Libellus in Italian, and that 
it was then translated into humanist Latin, probably by Mat- 
teo da Borgo 2 . In turn, Pacioli's version of it, published in 
1509, represents a re-translation into Italian, not Piero's ori- 
ginai text. In addition, although much of Piero's Libellus de- 
rives from his Trattato d'abaco, there are significant differen- 
ces between the two works. At the least they represent sep- 
arate phases in the development of his mathematics. Pos- 
sibly Pacioli pérceived this development. But, in his borrow- 
ings, Pacioli made no attempt to reconcile the variances the 
two manuscripts presented him. Although minimally he had 
to occupy himself with the translation of one of the man- 
uscripts, Pacioli seems to have been more interested in ex- 
panding the bulk of his literary oeuvre through a policy of 
editorial aggrandizement. 

The demonstration of these conclusions is dependent, in 
each case, upon detailed comparisons of texts and language, 
and sometimes of mathematical procedure. Some trusting read- 
ers may wish to spare themselves this material. 

The geometrical exercises of Piero's Latin Libellus are 
rendered in Italian, with only slight changes in numeration, 
at the end of Pacioli's Divina proportione. A comparison of 



2 See above, Chapter III, n. 8. 



Piero's Manuscripts and Pacioli's Adaptations 109 

the language used in the two treatises is instructive. Many 
Latin words or expressions that render fragmentary sentences 
whole, or sharpen syntax, or permit variety in diction, simply 
do not have correspondences in normal late Quattrocento 
written Italian. Thus, where Pacioli's text repeats and repeats 
the same limited fund of words, the Libellus, written in Latin 
for the Urbino court, can avail itself of a wealth of expressions 
to formulate the same ideas. Consequently its literary effect 
is of a much higher order. Some examples illustrate this point. 
Pacioli's resta is expressed in Latin as superit, as faciet reli- 
quum, as relinquitur, and as residuum est. Leva is rendered: 
aufert, removet, tollit; similarly, trailo: deducas, subtrahe, 
demas; piglia: cape, capitur, sume, sumatur. The ubiquitous 
fa of Pacioli's text is represented in a variety of expressions: 
reddentur, reddet, reddit, reddunt, agendum est, dabit, efficit, 
ejficietur, habebis, habemus, facit, fient, fuit, fit, fuerunt, 
eveniet and existat. Furthermore, Pacioli abbreviates in Italian 
many expressions that are given completely in Latin. 

Since the problems in Pacioli's « Particularis tractatus », 
in his Summa, derive from Piero's Abaco, and since the Abaco 
was, in turn, a preliminary step to Piero's Libellus, the cor- 
respondences among these works will bear examination. We 
may compare the Divina prò por t ione translation of Piero's 
Libellus both to the Trattato d'abaco and to the Latin Libel- 
lus to determine to which it is more closely related. If the 
Divina proportione tract appears dose to the Abaco text in 
similar problems, then it perhaps represents Piero's originai 
Italian text. On the other hand, if it is textually closer to the 
Latin Libellus, it may be considered a translation by Pacioli 
into Italian. 

The first comparison presents the case of a regular tetrahe- 
dron whose side is ^24 and whose axis is 4. What then is 
the length of a line drawn from an angle of the tetrahedron 
to the center of the axis. The three following textual examples 



HO Appendix Two 

(I, 1-3) show the Divina proportione « Libellus » to be closer 
to the Vatican Libellus in phrasing (and also in diction and 
method of mathematical solution) than to the Trattato d'abaco. 
This points to the conclusion that Pacioli's publication repre- 
sents a translation from the Latin rather than the originai 
Italian text of Piero. 



I, 1. Libellus de quinque corporibus, II, 6: 

jEst quadribasis triangularis aequilatera cuius latus est radix 
24, et axis est 4. Quantitas igitur quae a centro est ad quemli- 
bet angulorum reperienda est. 

Habes quadribasim ABCD, cuius quodlibet latus est radix 24: 
et axis AE est 4: et centrum F est in axe. Et quia est proportio 
ab AF ad AE, quae est a 3 ad 4, quae est proportio sexquiter- 
tia, erit AF 3/4 ipsius AE, qui est 4. Itaque AF est 3. Proba- 
tio. Dictum fuit quod unum ex lateribus est 24 et AE 3. Igitur 
FÉ est 1, quia AE est 4. Aufer AF, quae est 3: relinquitur 1 
FÉ. Et axis cadit super E, quae est 2/3 catheti BG. Et E est 
centrum basis BCD, et BG per eam quae praecessit est radix 
18. Cape 2/3, fiet radix 8. Trahe lineam BF, per penultimam 
primi Euclidis, potest quantum duae lineae BE, EF; et BF est 
3, et aequat AF, sicut per primam huius probatum fuit. Et 
BF potest 9, et EF potest 1, qui ablatus a 9 relinquetur 8, 
quae est vis BE, quae addita ad vim EF, quae est 1, efficit 9, 
et radix 9 est BF, quae est 3, et AF 3, CF 3, DF 3, nam omnes 
ortum habent a centro F, et in circumferentia terminantur. 



I, 2. Divina proportione, II, 5: 

Eglie uno quatro base triangulare equilatero che il suo lato e 
R .24. et laxis e .4. la quantità che dal centro a ciascuno angulo 
se vole trovare. 

Tu ai il quatro base .a.b.c.d. che ciascuno suo lato e R .24. Et 
laxis .a. e. e .4. et il centro i. e nel axis et per che quella pro- 
portione e da .ai. ad .a.e. che da .3. ad .4. che proportione 
sexquitertia sita .ai. trequarti de .a.e. che .4. adunqua .ai. e 
.3. aia prova e se dicto che uno di lati e R .24. et .ai. 3. dunqua 



Piero' s Manuscripts and Pacioli's Adaptations 111 

i.e. e .1. perche .a.e. e .4. tranne .ai. che .3. resta .1. f.e. et 
laxis cade sopra .e. che li doi terci del cateto .b.g. et .e. e centro 
dela basa .b.c.d. et .b.g. per la precedente e R .18. pigliane 
.2/3. sia R .8. tira la linea .bi. per la penultima del primo de 
Euclide pò quanto le doi linee .b.e. et .ei. et .bi. e .3. et e 
equale ad .ai. comme per la prima de questo fu provato et .bi . 
pò .9. et .ei. pò .1. trailo de .9. resta .8. che la posanca de 
.b.e. che gionta con la posanca de .ei. che .1. fa .9. et la R .9. 
e .bi. che .3. et .ai. 3, .ci. 3. d.f.3. per che tucte se puntano 
dal centro i. e terminano nela circumferentia. 



I, 3. Trattato d'abaco, fol. 105v (1): 

Egl'è una figura corporea che à quatro base triangulare de an- 
guli e lati equali, che ciascuno lato è radici de 24, et il suo assis 
è 4. Domando quanto è da ciascuno angulo al centro E. 
Tu ài la figura corporea de quatro base triangulare equilatera 
ABCD, et Passis suo è AE, et il centro è nell'assis in puncto 
F. Et perché AE è 4 et AF è sexquitertia ad AE, adunqua AF 
è 3. La prova. E s'è dicto che uno lato e radici de 24, però pigia 
la metà de radici de 24 ch'è radici de 6 et detrailo de 24, resta 
18, cioè radici de 18 è BH. E l'assis il quale cade in su li 
doi terci del diametro BH, ch'è radici de 18, et li 2/3 è radici 
de 8 che in sé montiplicato fa 8; et e' ss'è dicto che AF è 3, 
dunqua FÉ è 1. Perché AE è 4, tranne AF ch'è 3 resta FÉ, 1, 
che in sé multiplicato fa 1, gionto con 8 fa 9. Adunqua BF, CF, 
DF è ciascuno radici de 9, ch'è 3. 



A second set of comparisons (II, 1-3) reinforces the impres- 
sion created by the first. In this instance Piero treats the de- 
termination of the sides of a regular tetrahedron placed in a 
sphere whose axis is 7. Again, Pacioli's example in the Divina 
proportione appears as a translation from the Latin. 

II, 1. Libellus de quinque corporibus regularibus, II, 3: 

Quadribasis triangularis aequilateri et a spera contenti cuius 
diameter est 7, latus reperiendum. 



112 Appendix Two 

Per eam quae praecessit habes quod ea est proportio ab axe 
ad latus, quae est a latere ad diametrum sperae eum continentis. 
Habesque quod posse axis ad posse sui lateris est sexquialterum. 
Et similiter vis lateris ad vim diametri eadem proportione te- 
netur. Nunc habes diametrum, qui est 7, et posse suum est 49. 
Igitur posse diametri sperae est posse lateris quadribasis, et 
sicut 3 ad 2. Itaque die: si 3 esset 49, quod erit 2? Multiplices 
bis 49, habebis 98: partire per 3, eveniunt 32 2/3, cuius radix 
est latus quadribasis a spera contenti, et cuius diameter est 7. 



II, 2. Divina proportione, II, 2: 

Del quatro base triangolare equilatero contenuto dala spera che 
il suo diametro e .7. de lato suo investigare. 
Per la precedente ai che glie quella proportione dalaxis allato 
che e dal lato al diametro dela spera chel contene et ai che la 
posanca delaxis ala posanca del suo lato e sexquialtera et cosi 
quella dallato e al diametro hora tu ai il diametro che .7. e la 
sua posanca e .49. adunqua la posanca del diametro dela spera 
e la posanca del lato del quatro base si commo .3. ad .2. pero 
di se .3. russe .49. che seria 2. multiplica .2. via .49. fa .98. 
parti per 3 nevene .32 2/3. e la R .32 2/3. e ilato del quatro 
base contenuto dala spera che il suo diametro e 7. 



II, 3. Trattato d'abaco, fol. 107r (1): 

Egl'è uno corpo sperico che il suo diametro è 7; vogloci mectere 
dentro una figura de 4 base triangulare equilatera, che gPanguli 
contighino la circumferentia. Domando de' suoi lati. 
Devidi prima il diametro suo, ch'è 7, in tre parti equali che 
sirà ciascuna 2 1/3, montiplica 2 1/3 co' lo resto, ch'è 4 2/3, 
fa 10 8/9, il quale adoppia conmo radici fa radici da 43 5/9. 
De la quale radici fa 4 parti equali, cioè reca 4 a radici fa 16, 
parti 43 5/9 per 16; reca le parti a noni, arai 392 a partire 
per 144, ne vene 2 52/72, trailo de 43 5/9 resta radici de 24 
1/2 che è diametro de la basa del corpo triangulare. Et perché 
egl'è sexquitertia ad il lato de la basa, giogni 1/3 de 24 1/2, 
ch'è 8 1/16, ad 24 1/2 fa 32 2/3; et la radici de 32 2/3 sirà 
per il lato il dicto 4-base triangulare che intra nel corpo sperico 
che il suo diametro è 7 bracci. 



Piero' s Manuscripts and Pacioli's Adaptations 113 

That Pacioli dici not compare Piero's two treatises but 
adopted each independently is clear if the corresponding prob- 
lems in the Summa and the Divina proportione and the Li- 
bellus are compared, because of the textual relations that e* 
merge. The correspondences between the Divina proportione 
and the Libellus are far closer than those between the Summa 
and the Libellus. Two sets of examples illustrate this situation: 
III, 1-3 and IV, 1-3. 

Ili, 1. Libellus de quinque corporibus regularibus, II, 18: 

Sit cubus ABCDEFGH, habens latera quaeque ulnarum 4: quae- 
rendum est quot ulnarum erit quadratus? 
Dictum fuit in principio quadratomi», quod eorum quadratu- 
ra a suis lateribus habentur, si suum latus reduceretur ad cu- 
bum. Igitur multiplica suum latus in se, quod est 4, fiet 16, et 
quater 16 reddit 64. Dices ergo quod cubus ABCDEFGH, cuius 
latus sit 4, quadratus erit 64. 



Ili, 2. Divina proportione, II, 17: 

E se il cubo a.b.c.d.e.f.g.h. e per ciascuno lato .4. quanto sira 
quadrato se vole cercare. 

Fu dicto nel principio de quadrati che la sua quadratura favia 
dai suoi lati cioè recando il suo lato a cubo poi multiplica il 
suo lato che .4. in se fa .16. et .4. via .16. fa .64. adunqua 
dirai che il cubo ,a.b.c.d.e.f.g.h. che al suo lato .4. sia quadra- 
to .64. 



Ili, 3. Summa arithmetica, 6: 

Eglie un cubo .abcd.efgh. che per ciascun lato e .4. braccia, di- 
mando quanti braccia sira quadro. Moltiplica 4 in se fa .16. e 
poi .16. via .4. fa .64. e tanto e quadro facta. 



The relationships that already begin to be evident from the 



114 Appendix Two 

comparison of these three versions of a single problem can be 
traced in greater detail in the three redactions of the following 
problem (IV, 1-3). 

IV, 1. Libellus de quinque corporibus regularibus, II, 15: 

Sit latus cubi quadrilateri 4: quot erit diameter sperae, quae 
eum circumscribit investigandum est. 

Dico quod proportio possibilitatis diametri sperae ad eam quae 
est lateris cubi in illa descripta est tripla: hoc est sicut 3 ad 1. 
Ideo multiplica latus cubi in se, quod est 4, fiet 16. Die prae- 
terea. Si 1 esset 16, quot essent 3? Multiplica per 16, fiet 48, 
quem partire per 1, evenient 48: et 48 est vis diametri sperae 
quae cubum continet. Igitur diameter sperae est radix 48. Sed 
ut clarius intelligatur: habes cubum ABCDEFGH, extendas li- 
nearci AD, quae per penultimam primi Euclidis, potest quan- 
tum hae duae lineae AB et BD, quarum quaelibet est 4, et in 
se multiplicatae cum multiplicatione sua efficiunt 32. Itaque vis 
AD est 32. Et si trahas AH, eadem ratione, potest quantum 
duae lineae AD et DH, quae continent angulum D, qui rectus 
est. Et DH est 4, quae potest 16, et AD potest 32, qui additus 
ipso 16 efficit 48, quae est vis AH, quae linea transit per cen- 
trarci cubi et sperae. Et angulus A, et angulus H contingunt 
circumferentiam sperae. Igitur AH est diameter sperae, et sua 
vis est 48, et circumscribit cubum, cuius vis lateris est 16, quae 
est 1/3 possibilitatis diametri. 



IV, 2. Divina proportione, II, 14: 

Se illato del cubo equilatero e .4. che sira il diametro dela 
spera che il circumscrive investigare. 

Dico che la proportione dela posanca del diametro dela spera 
aquella dellato del cubo in quella descricto e tripla cioè comò 
.3. ad uno pero multiplica illato del cubo che .4. in se fa .16. 
hora di se uno fusse .16. eh saria .3. multiplica .3. via .16. fa 
.48. il quale parti per uno ne ven. 48. et .48. eia posanca del 
diametro dela spera che contene il cubo adunqua il diametro 
delaspera e R de .48. E per che meglio lo intenda tuai il cubo 
.a.b.c.d.e.f.g.h. tira la linea .a.d. laquale per la penultima del 
primo de Euclide pò quanto le do linee .a.b. et .b.d. che ciascu- 
na .4. che multiplicata ciascuna inse egionte insieme le multi- 



Piero' s Manuscripts and Pacioli's Adaptations 115 

plicationi fano .32. dunqua la posarla de .a.d. e .32. et se tu 
tiri .a.h. per quella medesima ragione pò quanto le do linee 
.a.d. et .d.h. che contengano langulo .d. che recto et .d.h. e .4. 
che pò .16. et .a.d. pò .32. che gionto con .16. fa .48. che la 
posanca de .a.h. la quale linea passa per lo centro del cubo e 
de la spera et langulo .a. e langulo .h. contingano la circum- 
ferentia dela spera adunqua .a.h. e diametro dela spera e la 
posanca sua e .48. et circumscrive il cubo che la posanca del 
suo lato e .16. eh .1/3. dela posan?a del diametro. 



IV, 3. Stimma arithmetica, 7: 

Eglie un cubo eh' per faccia .4. dimando che sia la diagonale 
interiore passante per lo centro K cioè laxis. Opera e troverai 
che sira la diagonale .ah. R .48. facta. 

If the corresponding problems in Piero's Trattato d'abaco 
(III, 4 and IV, 4) are joined to the two preceding sets of com- 
parisons, it is confirmed that Pacioli did not attempt to recon- 
cile the discrepancies that exist between Piero's texts. Instead, 
he translated or directly copied, of ten abbreviating the originals. 

Ili, 4. Trattato d'abaco, lo\. 106r(l): 

Adunqua egl'è uno cubo ABCD.EFGH che è, per ciascuno lato, 
4 bracci. Domando quanti bracci sirà quadrato. Tu di' ch'egl'è 
4 bracci per lato; dunqua montiplica 4 via 4 fa 16, et 4 via 16 
fa 64: però dirai ch'egl'è quadrato 64 bracci. 



IV, 4. Trattato d'abaco, fol. 106r (3): 

Egl'è uno cubo ch'è 4 per lato. Domando de la linea AH, pas- 
sante per lo cintro K, de Pangulo A a l'angulo H. 
Trova prima la linea diagonale AD che ài che è radici de 32, 
conmo per al precedente; poi montiplica DH, ch'è 4, in sé fa 16, 
giognilo con 32 fa 48 et radici de 48 sirà il diametro AH pa- 
sante per lo centro K. 



116 Appendix Two 

These examples demonstrate that the « Particularis tracta- 
tus » in Pacioli's Summa arithmetica derives exclusively from 
Piero's Trattato d'abaco, and, further, that the « Libellus » in 
Pacioli's Divina proportione is translated directly, and also 
exclusively, from Piero's Libellus in the Vatican 3 . 

A look at phrasing and mathematical procedure serves to 
confirm the above conclusions. Comparing, for instance, the 
problems concerning the measurement of a statue found in ali 
four treatises, we read as follows — Summa, 20: « Dico che tu 
faccia un vaso de legno... »; Abaco, fol. 109v (1): « Dico che 
tu facci uno vaso de legno... »; Divina proportione, IV, 17: 
« Fa uno vaso de legno ho d'altro... »; Libellus, IV, 17: « Con- 
ficias vas ex Ugno, aut ex alia re... ». The phrase, « aut ex alia 
re », found in both the Vatican Libellus and in the Divina pro- 
portione appears neither in the Trattato d'abaco nor in the 
Summa arithmetica. « Conficias » in the Vatican treatise and 
in the Divina proportione is the less succinct « Dico che tu 
facci » in the Summa and Trattato d'abaco 4 . 

This problem concerning the measurement of a statue con- 
firms, as well, that Pacioli did not draw upon both of his sour- 
ces in order to refine his method of mathematical solution. In 



3 One may further compare Summa, 4: Tu ai la figura corporea de A. 
base; Abaco, fol. 105v (1): « Tu ai la figura corporea de quattro base»-,. 
Libellus de quinque corporibus, II, 6: « Habes quadribasim ABCD »; Divina 
proportione, II, 5: « Tu ai il quatro base a.b.c.d. » Or, compare: Summa, 
6: « Montiplicata 4 in se fa 16 e poi 16 via 4 fa 64 e tanto e quadro facto; » 
and Abaco, fol. 106r (1): « Montiplicata 4 via 4 fa 16, et 4 via 16 fa 64: 
però dirai ch'egl'è quadrato 64 bracci. » These last two examples represent 
the full solutions. The Vatican Libellus, however, and following it, the 
Divina proportione « Libellus, » both contain a preface to this solution — 
Libellus, II, 18: « Dictum fuit in principio quadratorum, quod eorum quadra- 
tura a suis lateribus habentur, si suum latus reduceretur ad cubum »; Divina 
proportione, II, 17: « Fu dicto nel principio de quadrati eh' la sua quadratura 
favia da suoi lati cioè recando il suo lato a cubo. » 

4 The problem is reproduced in full above, from the Summa arithme- 
tica, in Appendix I, pp. 104-05. 



Piero 1 's Manuscripts and Pacioli's Adaptations 117 

the Vatican Libellus, one marks the height o£ the water in a 
vat, puts the statue into the vat, waits for the water to become 
stili, makes a new mark for the higher water level, removes 
the statue, and measures the distance between the first and sec- 
ond marks. The method is identical in the Divina proporzione. 
In the Trattato d'abaco, however, one marks the water level, 
adds the statue, marks the higher water level, and measures the 
distance between the first and second levels. In the Summa 
arithmetica the same process is demonstrated. The phrase, « et 
sinas aquam quietam stare, » in the Vatican Libellus, and « e 
lassa reposare l'aqua », in the Divina proportione, occurs in nei- 
ther the Trattato d'abaco nor in the Summa arithmetica. Nor 
is the statue removed from the vat in the latter two treati- 
ses 5 . 

The wording, and phrasing, of identical problems in the 
Summa and in the Divina proportione is at times widely vari- 
ant. The sentence constructions, however, relate closely to 
the originai sources in Piero's manuscripts. While Piero's 
Italian is simple, at times even elementary, Pacioli's Italian 



5 We read in the Summa, 20: « Poi fa un segno dove agiogni laqua e 
mettivi dentro la statua che tu voi mesurare e vedi quanto e cresciuta 
laqua e fa unaltro segno e sommo laqua ritto a quello de prima. Poi mesura 
dal primo segno al secondo e vedi quanto eglie; » and similarly in the 
Abaco, fol. 109v (1): « Poi poni o fa' uno segno a sommo l'aqua et poi ne 
mecti dentro la statua che tu voi mesurare e vedi quanto è chrescuta l'aqua 
e fa' un altro segno a sommo l'aqua, derido al segno de prima. Poi mesura 
dal primo segno al secondo e vedi quanto egl'è. » On the other hand, the 
Divina proportione, IV, 17: « Poi fa uno segno nel vaso a sommo laqua 
et poi mecti dentro la statua che tu voi mesurare e lassa reposare l'aqua 
poi vedi quanto e cresciuta et fa a sommo laqua un altro segno derido a 
quello de prima poi trafora la statua et mesura quanto e dal primo segno 
al secondo,-» and the Libellus de quinque corporibus, IV, 17: «Deinde 
signa eam partem vasis, quae a summitate aquae tangitur. Postea iustus 
ponas statuam quam vis metiri, et sinas aquam quietam stare. Deinde videas 
quantum creverit, et facias aliud signum ad summitatem aquae de directo 
supra primum, deinde extrahe statuam, et mensura quantum distat secun- 
dum signum a primo. » 



11° Appendix Two 

in the Divina proportione is distinguished by greater concision, 
clarity, and syntactical rigor. These differences, together with 
the discrepancies in content, suggest that the Divina proportio- 
ne « Libellus » (despite the simplifications pointed out) is a 
translation into Italian from a concise humanist Latin text. 
One perhaps must conclude that Piero's originai Italian man- 
uscript for the Vatican Libellus had much the same character 
as the problems in the Trattato d'abaco, which are written in 
a rough, unpolished style, and that the translator was almost 
entirely responsible for the literary ornaments of the Latin 
text. 



Table I 



Derivations of Exercises in the «Summa arithmetica» from the 
« Trattato d'abaco » 



Summa arithmetica ... 
Part II, fols. 68 v -73 v 
problem no.: 

2 

3 

4 

5 

6 

7 

8. 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 



Ash 


DurnliÉ 


tm 280 


Trattato d'abaco 


fol. 


105 r 


(2) 


fol. 


105 r 


(3) 


fol. 


105 v 


(1) 


fol. 


105 v 


(2) 


fol. 


106 r 


(1) 


fol. 


106 r 


(2) 


fol. 


106 r 


(3) 


fol. 


106 r 


(4) 


fol. 


106 v 


(1) 


fol. 


106 v 


(2) 


fol. 


106 v 


(3) 


fol. 


107 r 


(1) 


fol. 


107 r 


(2) 


fol. 


107 r 


(3) 


fol. 


107 v 


(1) 


fol. 


108 r 


(1) 


fol. 


108 v 


(1) 


fol. 


109 r 


(1) 


fol. 


109 v 


(1) 


fol. 


110 r 


(1) 


fol. 


110 r 


(2) 


fol. 


110 v 


(1) 


fol. 


lll r 


(1) 


fol. 


112 r 


(1) 


fol. 


112 v 


(1) 



120 Table I 

Summa arithmetica ... Ashburnham 280: 

Part II, fols. 68 v -73 v Trattato d'abaco 
problem no.: 

27 fol. 113 r (1) 

28 fol. 113 r (2) 

29 fol. 113 r (3) 

30 fol. 113 v (1) 

3 1 

32 fol. 114 r (1) 

33 fol. 114 r (2) 

34 fol. 114 r (3) 

— . ----- fol. 114 r (4) 

35 fol. 114 v (1) 

36 fol. 115 r (1) 

37 fol. 115 r (2) 

38 fol. 115 v (1) 

39 fol. 115 v (2) 

40 fol. 115 v (3) 

41 - fol. 116 r (1) 

42 _ fol. 116 v (1) 

43 fol. 116 v (2) 

44 - fol. 117 r (1) 

45 fol. 117 r (2) 

46 fol. 117 r (3) 

47 fol. 117 r (4) 

48 fol. 117 v (1) 

49 fol. 117 v (2) 

50 fol. 117 v (3) 

51 fol. 118 r (1) 

52 fol. 118 v (1) 

53 1 fol. 118 v (2) 

53 2 fol. 119 r (1) 

54 fol. 119 r (2) 

55 fol. 119 v (1) 

56 fol. 119 v (2) 



Table II 



Derivations of Exercises in the « Libellus de quinque 
corporìbus regularibus » from the « Trattato d'abaco » 



Libellus de quinque corporibus 
regularibus (Vat.Urb.lat. 632) 

Part I: 

3. -, 

4. , 

8. 

11 

14 

15 

16 

17 

18 

19 

20 - 

21 

22 

23. - 

24 

25 

28 

29 

30 

34. - 

35 

36 

37 

38 



Trattato d'abaco (Bibl. 


Laurenziana, Ashb. 280) 


fol. 


80 v 


(1) 


fol. 


79 r 


(1) 


fol. 


100 r 


(1) 


fol. 


101 r 


(2) 


fol. 


83 r 


(1) 


fol. 


83 r 


(2) 


fol. 


83 v 


(1) 


fol. 


83 v 


(2) 


fol. 


83 v 


(3) 


fol. 


84 r 


(1) 


fol. 


84 r 


(2) 


fol. 


84 r 


(3) 


fol. 


102 r 


(1) 


fol. 


102 v 


(1) 


fol. 


103 v 


(1) 


fol. 


103 r 


(2) 


fol. 


90 r 


(1) 


fol. 


88 v 


(1) 


fol. 


88 v 


(2) 


fol. 


89 r 


(1) 


fol. 


89 r 


(2) 


fol. 


90 r 


(1) 


fol. 


91 v 


(1) 


fol. 


91 v 


(2) 



122 



Table II 



Libellus de quinque corporibus 
regularibus (Vat.Urb.lat. 632) 



Trattato d'abaco (Bibl. 
Laurenziana, Ashb. 280) 



39. 
40. 
42. 
43. 
44. 
45. 
46. 
47. 
49. 



4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
18. 
21. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 



Part II: 



fol. 


92 r 


(1) 


fol. 


92 v 


(1) 


fol. 


92 v 


(2) 


fol. 


93 r 


(1) 


fol. 


93 r 


(2) 


fol. 


93 v 


(1) 


fol. 


93 v 


(2) 


fol. 


93 v 


(3) 


fol. 


94 r 


[D 


fol. 


105 r 1 


1) 


fol. 


105 r < 


3) 


fol. 


105 v 


(1) 


fol. 


116 r 


(1) 


fol. 


115 r 


r 2) 


fol. 


115 v 


(1) 


fol. 


115 v 


(2) 


fol. 


115 r 


(1) 


fol. 


116 v 


(1) 


fol. 


116 v 


(2) 


fol. 


117 r 


r D 


fol. 


106 r 


(3) 


fol. 


107 r 


r 2) 


fol. 


106 r 


r D 


fol 


114 r ( 


1) 


fol. 


114 v 


(1) 


fol. 


110 r 


[2) 


fol. 


110 r ( 


1) 


fol. 


110 v 


(1) 


fol. 


lll r 


1) 


fol. 


112 r 


1) 


fol. 


112 v 


(1) 


fol. 


113 r 


r D 


fol. 


113 r 


r 2) 


fol. 


113 r 


r 3) 


fol. 


113 v 


(1) 



Table II 123 

Libellus de quinque corporibus Trattato d 'abaco • (Bibl. 

regularibus (Vat.Urb.lat. 632) Laurenziana, Ashb. 280) 

Part III: 

15 fol. 

16 f: 

20 fo1 - 

21 - *f 

22 — *°1. 

23'. "".::.:..: foi. 

24 °- 

25 °- 

27 foL 

28 fo1 - 

29 fo1 - 

Part IV: 

17 foL 



106 v 


(2) 


106 v 


(3) 


117 r 


(3) 


117 v 


(1) 


117 r 


(4) 


117 v 


(3) 


117 v 


(2) 


118 r 


(1) 


118 v 


(1) 


118 v 


(2) 


119 r 


(2) 


119 v 


(1) 


119 v 


(2) 


109 r 


(1) 


109 v 


(1) 



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Illustrations 



1. Column and barre]. MS. 
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£oì, 61r. 

6. Mazzocchio, Piero, Prospettiva pingendt. 





7. Mazzocchio. Lorenzo da Lendinara, Duomo, Modena. 



8. «Un pozzo de sei faccie». Piero, "Prospectiva pingendi. 





9. Hexagonal well. Lorenzo da Lendinara, Duomo, Modena. 



10. «Tempio de octo facce», Piero, Prospectiva pìngenàì. 

-ti 



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II. Octagonal Baptistery. Lorenzo da Lendinara, Duomo, Modena. 



12. «Corpus 72 basium». Piero, Libelius de quinque corportbus, Col. 
51r. 



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13. «Una cupola... commo uno quarto de una pala». Piero, Prospec- 

tiva pingcndi. 



14. Intarsia apsc. Gio- S 
vanni da Verona,. 
Monteoliveto ( Sie- 
na). 





15. Portrait of Luca Padoli, Jacopo de' Barbari, Museo di Capodi- 
monte, Naples. 




16, Pomaù of Luca Pacioli. 
L. Paridi, Stimma de 
arithmetìca^ fol. lllv. 



17, Icosahedron. Piero, 
Trattato d'abaco, 
foL 113v. 






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18. Dodecahedron. Pie- 
ro, Libellus de 
quinque corport- 
btts, fol. 33v. 



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20. Icosahexahedron. L. 
Pacioli, Divina prò- 
portione. 




21. Seventy-two faced 
body. L. Paciolì, 
Divina proportione. 



22. Stereometrìe bod- 
ies. Giovanni da 
Verona, S. Maria 
dell'Organo, Vero- 



ni. 




23. Pattern for dode- 
eahedron. A, DQ- 
rer, Unterweisung 
der Messuftg. 







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24, Cube in perspective. A, Durer, Unterweisung der Messung. 



25. Foreshortened head, A. Durer, Proporlionslebre. 









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27. Capital study. Piero, Prospecitva pingendi. 



28. Geometricized human figure. 
A, Dùrer, Proporlìonslehre, 




s « 



29. Geometricized human figure. 
Ai Diirer, Dresden Skctch- 
book. 



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31. Irregular solids. W. Jamnitzcr, Perspectiva corporum regularìum. 



32. Dodecahedron. D, Barbaro, La 
pratica della perspeitiva. 





33. Capitai study. D. Bar- 
baro, La pratica della 

perspettiva. 



m 34. The live regular botl- 
ies. D. Barbaro, «La 
pratica della prospet- 
tiva» . 







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35. Stellateci body. D. Barbaro, « La pratica della prospettiva ». 



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Indexes 



List of Illustrations 



1. Column and barrel. Florence, Biblioteca Medicea Lau- 
renziana, MS. Cod. Acq. e doni 154, fol. 218r . . . p. 139 

2. Well. Florence, Biblioteca Medicea Laurenziana, MS. 

Cod. Acq. e doni 154, fol. 220r » 139 

3. « Palla di S. Liperata ». Florence, Biblioteca Laurenzia- 
na, MS. Cod. Acq. e doni 154, fol. 219v » 140 

4. « Una volta di crociera ». Piero, Vrospectiva pingendi » 140 

5. « Volta per modum crucis ». Piero, Libellus de quinque 
corporibus regularibus, fol. 61r » 141 

6. Mazzocchio. Piero, Pros peci iva pingendi . . . , » 141 

7. Mazzocchio. Lorenzo da Lendinara, Duomo, Modena . » 142 

8. « Un pozzo de sei f accie ». Piero, Vrospectiva pingendi . » 142 

9. Hexagonal well. Lorenzo da Lendinara, Duomo, Modena » 143 

10. « Tempio de octo facce. » Piero, Vrospectiva pingendi , » 143 

11. Octagonal Baptistery. Lorenzo da Lendinara, Duomo, 
Modena » 144 

12. « Corpus 72 basium ». Piero, Libellus de quinque cor- 
poribus regularibus, fol. 5 Ir » 144 

13. « Una cupola ... commo uno quarto de una pala ». Piero, 
Vrospectiva pingendi » 145 

14. Intarsia apse. Giovanni da Verona, Monteoliveto (Siena) » 145 

15. Portrait of Luca Pacioli. Jacopo de' Barbari, Museo di 
Capodimonte, Naples . . . . . . . • » 146 



162 List of lllustrations 

16. Portrait of Luca Pacioli. L. Pacioli, Stimma de arithme- 

tica, fol. lllv » 146 

17. Icosahedron. Piero, Trattato d'abaco, fol. 113v ...» 147 

18. Dodecahedron. Piero, Libellus de quìnque corporibus, 

fol. 33v » 147 

19. Irregular body. Piero, Trattato d'abaco, fol. 108r. . . » 147 

20. Icosahexahedron. L. Pacioli, Divina proportione . . » 148 

21. Seventy-two faced body. L. Pacioli, Divina proportione » 148 

22. Stereometrie bodies. Giovanni da Verona, S. Maria del- 
l'Organo, Verona » 149 

23. Pattern for dodecahedron. A. Dùrer, Unterweisung der 
Messung » 149 

24. Cube in perspective. A. Dùrer, Unterweisung der 
Messung » 150 

25. Foreshortened Head. A. Dùrer, Proportionslehre . . » 150 

26. Head study. Piero, Prospectiva pingendi . . . . » 151 

27. Capital study. Piero, Vrospectiva pingendi . . . . » 152 

28. Geometricized human figure. A. Dùrer, Proportionslehre » 153 

29. Geometricized human figure. A. Dùrer, Dresden 
Sketehbook J » » 153 

30. Geometricized human figure. A. Dùrer, Dresden 
Sketehbook » 154 

31. Irregular solids. W. Jamnitzer, Perspectiva corporum 
regularium » 155 

32. Dodecahedron. D. Barbaro, La pratica della perspettiva » 155 

33. Capital study. D. Barbaro, La pratica della perspettiva » 156 

34. The five regular bodies. D. Barbaro, «La pratica della 
prospettiva » » 156 

35. Stellated body. D. Barbaro, «La pratica della prospettiva» » 157 

36. Irregular mazzocchio. D. Barbaro, « La pratica della pro- 
spettiva » » 158 



Contents 

Riassunto in lingua italiana P- VII 

I. Artists and Mathematicians : Arithmetical, Geometrical 

and Perspectival Studies » 1 

II. The «Trattato d'abaco»: Algebra and Geometry . . » 21 

III. The « Libellus de quinque corporibus regularibus »: 

Regular and Irregular Bodies » 44 

IV. Applications of the Art of Measurement ...» 64 

Appendix I. Luca Pacioli and the « Trattato d'abaco » . » 98 

Appendix II. Piero's Manuscripts and Pacioli's Adaptations: 

Textual Questions » 107 

Table I. Derivations of Exercises in the « Summa arithme- 

tica » from the «Trattato d'abaco» . . . . » 119 

Table II. Derivations of Exercises in the « Libellus de quin- 
que corporibus regularibus » from the « Trattato 
d'abaco» » 121 

Bibliography » 124 

Illustrations » 137 



Finito di stampare 

nel mese di ottobre 1977 

dalla Tipografia Leonelli 

di Villanova di Castenaso 

per A. Longo Editore 

in Ravenna 



This volume is a study in the relationship 
between Renaissance mathematica! theory and 
artistic practìce. The author examines die 
specific case of Piero, the artist and mathema- 
tician, whose paìntings are today acclaimed abo- 
ve ali others for their geometrie perfection. Al* 
though hìs treatise on perspective is discussed 
whenever perspective is the order of the day, 
Piero's two purely mathematica! works, tlie 
Trattato d'aboco and the Libellus de quinque 
corporibus regularìbus, have been neglected by 
historians o£ art, It is, however, througb these 
two «. minor » treatise that Piero 's mathema- 
tical abilities, highly celebrated in bis own 
time, are brought once again into focus. The 
author investigates these works in dose detail, 
tracing both their sources in classical and late- 
medieval mathematics and the internai develop- 
ment of Piero's mathematics that they reveal, 
as well as their relationships with Piero's 
perspective treatise. Following the esposition 
of the content of these works, the study ana- 
lyzes the ìmplicatìons of Piero's ideas and their 
diffusion in the realm of theory and practìce 
of the figurative arts and architecture. 



MARGARET DALY DAVIS has studied the 
history of art in America and Europe. A 
graduate of New York University, she received 
the Ph, D. from the University of North Ca- 
rolina at Chapel Hill. In recent years she has 
been living in Florence. 



L. 8.000 
(7J47) 



Speculum Artium 

Collana diretta da Aldo Scaglione 



Questa collana intende mettere alla prova la validità del 
metodo interdisciplinare nella storia della cultura, sia nel 
caso di probemi specifici che per investigazioni teoriche 
sulla natura di tale metodo. Comprenderà quindi esempi 
delle proposte più aggiornate concernenti i rapporti fra 
le varie arti, fra arte e pensiero, fra arte e scienza, fra le 
arti e la storia sociale, e fra le arti letterarie o figurative 
e le tecniche che ad esse sottostanno, dalle tradizionali 
arti del Trivio e Quadrivio alle convenzioni formali quali 
la metrica, gli stili, le forme dei generi, l'iconografia. 
Testi e saggi di qualsiasi area della cultura occidentale 
appariranno nella lingua originale dell'autore. 



This Series intends to test the vaHdity of mterdisciplinary 
approaches in cultural history both as applied to spedile 
questiona and as theoretical exploration of the implicatìons 
of such methods. It wìlì therefore include responses to 
the most up-to-date desiderata concerning the relationships 
between the various atts, between art and ideas, between 
art and science, between the arts and social history, and 
between the literary or figurative arts and the techniques 
underlying them, from the traditional arts of the Trivium 
and Quadrivium to such formai conventions as metrics, 
styles, genre formats, and iconogiaphy. 
Texts and essays from any area of western culture wUl 
appear in the originai language of the author.